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Qcil  &  Mechanical  Engineer. 

BAN  FBAKGISOO,  GAL. 

ELEMENTS 


ANALYTICAL    ]\fECeANI(: 


S 

'^^N 


W.   li.   C.   BAETLETT,   LL.D., 

PROFESSOR  OF    NATURAL  AND    EXPERIMENTAL    I'lnLOSOPHY    IN    TFIE    U\ITE"II 
STATES  I\IILITAUY  ACADEMY  AT   WEf!T    TOIXT. 

AND 

AUTHOR  OF  ELEMENTS  OF  SYNTHETICAL  MECHANICS,  ACOUSTICS, 
OPTICS,  AND  SPHERICAL  ASTRONOMY. 


SEVENTH  EDITION,  REVISED,  CORRECTED,  AND  ENLARGED. 


NEW   YORK: 
PUBLISHED    BY    A.   S.    BARNES    &    BURR, 

51   &  53   JOHN-STREET. 
1860. 


Entei'ed  accordine'  to  Act  of  Con:;rcss.  iu  the  year  One  ThousaDd 

Eight  Hundred  and  Fifty-eight, 

By     W.     II.    C.     BART  LETT, 

In  the  Clerk's  Office  of  the  District  Court  of  the    United    States  for   the  Southern 
District  of  New-York. 


O  .    W  .    ".V  ODD, 

PrhiUr, 
John-street,  cor.  Dutch. 


Library 


TO 

COLONEL  SYLVAiNUS  THAYER, 

OF  THE  OOEPS  OF  ENGINEERS,  AND  LATE  SUPEKINTENDEXT  OF  THE 

UNITED  STATES   MILITAItY   ACADEMY, 
IS  MOST  EESPECTFULLY  AND   AFFECTIONATELY  DEDICATKU, 

IN    GHATITUDE    FOR    THE    PRIVILEGES 

ITS    ATJTnOR    HAS    ENJOYED    UNDER    A    SYSTEM    OF    INSTRUCTION 

AND    GOVERNMENT    WHICH    GAA'E    VITALITY    TO 

THE  ACADEMY, 

AND    OF    WHICH    HE    IS    THE    FATHEK, 


733004 


PREFACE. 


It  is  now  six  years  since  the  publication  of  the  first  edi- 
tion of  the  present  work.  During  this  interval,  it  has  l)een 
corrected  and  amended  according  to  the  suggestions  of  daily 
experience  in  its  nse  as  a  text-book.  It  now  appears  with  an 
additional  part,  nnder  the  head.  Mechanics  or  Molecules  ;  and 
this  completes — in  so  far  as  he  may  have  succeeded  in  its  ex- 
ecution— the  design  of  the  author  to  give  to  the  classes  com- 
mitted to  his  instruction,  in  the  Military  Academy,  what  has 
ajjpeared  to  him  a  proper  elementary  basis  for  a  systematic 
study  of  the  laws  of  matter.  The  subject  is  the  action  of 
forces  upon  bodies, — the  source  of  all  })hysical  phenomena — and 
of  which  the  sole  and  sufficient  foundation  is  the  comprehensive 
fact,  that  all  action  is  ever  accompanied  by  an  ecpial,  contrary, 
and  simultaneous  reaction,  l^either  can  have  precedence  of 
the  other  in  the  order  of  time,  and  from  this  comes  that  char- 
acter of  permanence,  in  the  midst  of  endless  variety,  apparcn; 
in  the  order  of  nature.  A  mathematical  formula  which  sliall 
express  the  laws  of  this  antagonism  will  contain  the  whole  sub- 
ject; and  whatever  of  specialty  may  mark  our  perceptions  of 
a  particular  instance,  will  be  found  to  have  its  origin  in  corre- 
sponding peculiarities  of  physical  condition,  distimce,  i[>lace, 
and  time,  which  are  the  elements  of  this  formula.  Its  discus- 
sion constitutes  the  study  of  Mechanics.  All  phenomena  in 
which  bodies  have  a  part  are  its  legitimate  subjects,  and  no 
form  of  matter  undei-  extraneous  influences  is  exempt  from  its 


iv  PEEFACE. 

scrutiny.  It  emljraces  alike,  in  tlieir  reciprocal  action,  tlie 
gigantic  and  distant  orbs  of  the  celestial  regions,  and  the 
proximate  atoms  of  the  ethereal  atmosphere  which  pervades 
all  space  and  establishes  an  mibroken  continnity  upon  which 
its  Divine  Architect  and.  Anthor  may  impress  the  power  of 
His  will  at  a  single  point  and  be  felt  everywhere.  Astronomy, 
terrestrial  physics,  and  chemistry  are  but  its  specialties ;  it 
classifies  all  of  human  knowledge  that  relates  to  inert  matter 
into  groups  of  phenomena,  of  which  the  rationale  is  in  a  com- 
mon principle ;  and  in  tlie  hands  of  those  gifted  with  the 
priceless  boon  of  a  copious  mathematics,  it  is  a  key  to  exter- 
nal nature. 

The  order  of  treatment  is  indicated  by  the  heads  of  Me- 
chanics OF  Solids,  of  Fluids,  and  of  Molecules, — an  order  sug- 
gested by  differences  of  physical  constitution. 

The  author  would  acknowledge  his  obligation  to  the  works 
of  many  eminent  writers,  and  particularly  to  those  of  M.  La- 
grange, M.  Poisson,  M.  Couchey,  M.  Fresnel,  M.  Lame,  Sir 
William  E.  Hamilton,  the  Eev.  Baden  Powell,  Mr.  Airy,  Mr. 
Pratt,  and  Mr.  A.  Smith. 

West  Poixt,  1858. 


coxte:^ts. 


INTRODUCTION. 

PAGB. 

Preliminary  Definitions H 

Physics  of  Ponderable  bodies H 

Primary  Properties  of  Bodies 15 

Secondary  Properties 1*5 

Force 20 

Pliysical  Constitution  of  Bodies 22 

PAKT   I. 

MECHANICS    OF    SOLIDS. 

Space,  Time,  Motion,  and  Force 31 

Work 38 

Varied  Motion , 42 

Equilibrium 46 

The  Cord 47 

The  Muffle 48 

Equilibrium  of  a  Pugid  System — Virtual  Velocities 50 

Principle  of  D'Alembert 55 

Free  Motion 58 

Composition  and  Resolution  of  Oblique  Forces G2 

Composition  and  Resolution  of  Parallel  Forces 75 

Work  of  Resultant  and  of  Component  Forces 82 

Moments 84 

Composition  and  Resolution  of  Moments SS 

Translation  of  General  Equations 01 

Centre  of  Gravity 98 

Centre  of  Gravity  of  Lines 97 

Centre  of  Gravity  of  Surfaces ■  102 

Centre  of  Gravity  of  Volumes 109 

Centrobaryc  Method 114 

Centre  of  Inertia .• 116 

Motion  of  the  Ceutve  of  Inertia 118 


n  CONTENTS. 

PAOB. 

Motion  of  Tianslatioii 120 

Gcnoral  Theorem  of  Work  and  Living  Force  .  , 120 

Stable  and  Unstable  Equilibrium 123 

Initial  Conditions,  Direct  and  Reverse  Problem 126 

Vertical  Motion  of  Heavy  Bodies 127 

Projectiles 135 

Piotary  Motion 105 

Moment  of  Inertia,  Centre  and  Kadiu.s  of  Gyration 175 

Impulsive  Forces 169 

]\Iotion  under  the  Action  of  Impulsive  Forces 171 

Motion  of  the  Centre  of  Inertia 171 

Motion  about  the  Centre  of  Inertia 173 

Angular  Velocity 17-1 

Motion  of  a  System  of  Bodies 179 

Motion  of  Centre  of  Inertia  of  a  System 1  SO 

Motion  of  the  Sj-stem  about  its  Centre  of  Inertia 181 

Conservation  of  the  Motion  of  tiie  Centre  of  Inertia  of  a  System 181 

Conservation  of  Areas 183 

Invariable  Plane 185 

Principle  of  Living  Force 186 

Planetary  Motions 188 

Laws  of  Central  Forces 190 

Orbits l-'G 

System  of  the  World 198 

Consequences  of  Kepler's  Laws 198 

Perturbations  ' ^ . . .  203 

Coexistence  and  Superposition  of  Small  Motions 205 

Universal  Gravitation 20G 

Impact  of  Bodies ^ 211 

Constrained  Motion  on  a  Surface 218 

on  a  Curve 220 

"                 "         about  a  Fixed  Point 246 

about  a  Fixed  Axis 247 

Compound  Pendulum 249 

Ballistic  Pendulum 259 


PART    II. 

MECHANICS    OF    FLUIDS. 

Introductory  Remarks 263 

Mariotte's  Law 265 

Law  of  Pressure,  Densitj',  and  Temperature 266 

E(iu:il  Transmission  of  Pressure 268 

iilotion  of  Fluid  Particles 270 

Equilibrium  of  Fluids 280 

Pressure  of  Heavy  Fluids 289 

Equilibrium  and  Stability  of  Floating  Bodies 295 


CONTENTS.  Vll 

PAGE. 

Specific  Gravity "U4 

Atmosplieiic  Pressure -jI  ^j 

Barometer 317 

Motion  of  Heavy  Incompressible  Fluids  in  Vessels o20 

Motion  of  Elastic  Fluids  in  Vessels 338 


PAET    III. 

MECHANICS     OF    MOLECULES. 

Introductory  Remarks 345 

Periodicity  of  Molecular  Condition 345 

Waves 352 

Wave  Function 353 

Wave  Telocity 3G0 

Relation  of  Wave  Velocity  to  Wave  Length 3G3 

Surface  of  Elasticity 305 

Wave  Surface 367 

Double  Wave  Velocity 372 

Umbilic  Points 375 

Molecular  Orbits 378 

Reflexion  and  Refraction 381 

Picsolution  of  Living  Force  by  Deviating  Surfaces 384 

Polarization  by  Reflexion  and  Refraction 388 

Diffusion  and  Decay  of  Living  Force 394 

Interference •  •  •  395 

Inflexion 400 

PART    lY. 

APPLICATIONS    TO    SIMPLE    MACHINES,    PUMPS,    &c. 

Genera]  Principles  of  all  Machines 405 

Friction 407 

Stiffness  of  Cordage 415 

Friction  on  Pivots 420 

Friction  on  Trunnions ...  425 

Tiie  Cord  as  a  Simple  Machine 429 

The  Catenary 439 

Friction  between  Cords  and  Cylindrical  Solids 441 

Inclined  Plane 443 

The  Lever 44G 

Wheel  and  Axle 449 

Fixed  Pulley 451 

^lovable  Pulley 454 

The  Wedge 4G0 

The  Screw : 404 


Vlll  CONTENTS. 

PAGE. 

Pumps 4G9 

The  Siphon 479 

The  Air-Pump 481 


TABLES. 

Table    T. — The  Tenacities  of  Different  Substances,  and  the  Resistances  wliicli 

they  oppose  to  Direct  Compression 488 

"  II. — Of  the  Densities  and  Volumes  of  Water  at  Different  Degrees  of 
Heat  (according  to  Stampfer),  for  every  2^  Degrees  of  Fah- 
renheit's Scale 490 

"     III. — Of  the  Specific  Gravities  of  some  of  the  most  Important  Bodies.  491 

"     IV. — Table  for  finding  Altitudes  with  the  Barometer 494 

"       V. — Coefficient  Values,  for  the  Discharge  of  Fluids  through  thin  Plates, 

the  Orifices  being  Remote  from  the  Lateral  Faces  of  the  Vessel .  49(3 

"     VI. — Experiments  on  Friction,  without  Unguents.     By  M.  Morin 497 

"    VII. — Experiments  on  Friction  of  Unctuous  Surfaces.     By  M.  Morin  ....  500 

"  VIII. — Experiments  on  Friction  with  Unguents  interposed.    By  M.  Morin.  501 

"     IX. — Friction  of  Trunnions  in  their  Boxes 503 

"       X. — Of  Weights  necessary  to  Bend  different  Ropes  around  a  Wheel 

one  Foot  in  Diameter .  504 


The  Greek  Ali:ihabet  is  here  inserted  to  aid  those  who  are  not  already  famil- 
iar with  it,  in  reading  the  parts  of  the  text  in  which  its  letters  occur. 


Letters. 


A  a 

Alpha 

B  /3? 

Beta 

r  yf 

Gamma 

A   6 

Delta 

E  s 

Epsilou 

z?C 

Zeta 

Ht] 

Eta 

0  9,6 

Theta 

I    1 

lota 

K  X 

Kappa 

A  X 

Lambda 

Mf/. 

Mu 

N  V 

Nu 

S  g 

Xi 

O  0 

0  micron 

IT  TO'-n- 

Pi 

P  PS 

■Rho 

2   (fg 

Sigma 

T  t7 

Tau 

T  V 

Upsilon 

<t>  9 

Phi 

x% 

Chi 

Y4. 

Psi 

n  w 

Omega 

ELEMENTS 


ANALYTICAL  3IECIIANICS. 


INTEODUCTIOX. 


The  term  nature  is  employed  to  signify  the  assemblage  of  all 
the  bodies  of  the  miiverse ;  it  includes  whatever  exists  and 
is  the  subject  of  change.  Of  the  existence  of  bodies  we  are 
rendered  conscious  by  the  impressions  they  make  on  our  senses. 
Their  condition  is  subject  to  a  variety  of  changes,  whence  we 
infer  that  external  causes  are  in  oj^eration  to  produce  them  ;  and 
to  investigate  nature  with  reference  to  these  changes  and  their 
causes,  is  the  object  of  Physical  Science. 

All  bodies  may  be  distributed  into  three  classes,  viz :  unorgan- 
ized or  inanimate.^  organized  or  animated^  and  the  heavenly  todies 
OY  primary  organizations. 

The  unorganized  or  inanimate  bodies,  as  minerals,  water,  air, 
form  the  lowest  class,  and  are,  so  to  speak,  the  substratum  for  the 
others.  These  bodies  are  acted  on  solely  by  causes  external  to 
themselves  ;  they  have  no  definite  or  periodical  duration  ;  nothing 
that  can  properly  be  termed  life. 

The  organized  or  animated  bodies,  are  more  or  less  perfect 
individuals,  possessing  organs  adapted  to  the  performance  of  cer- 
tain appropriate  functions.  In  consequence  of  an  innate  principle 


12  ELEMENTS     OF     ANALYTICAL     MECHANICS 

peculiar  to  them,  known  as  vitaliUj^  bodies  of  this  class  are  con- 
stantly ai)]jropriating  to  themselves  unorgani/ecl  matter,  changing 
its  properties,  and  deriving,  by  means  of  this  process,  an  increase 
of  bvilk.  They  also  posse^is  the  faculty  of  reproduction.  They 
retain  only  for  a  limited  time  the  vital  principle,  and,  when  life 
is  extinct,  they  sink  into  the  class  of  inanimate  bodies.  The 
animal  and  vegetable  kingdoms  include  ail  the  species  of  this 
class  on  oui'  earth. 

The  celestial  hodies,  as  the  fixed  stars,  the  sun,  the  comets, 
plaiiets  and  their  secondaries,  are  the  gigantic  individuals  of  the 
universe,  endowed  with  an  organization  on  the  grandest  scale. 
Their  constituent  j^arts  may  be  compared  to  the  organs  possessed 
by  Ijodies  of  the  second  class ;  those  of  our  earth  are  its  conti- 
nents, its  ocean,  its  atmospliere,  which  are  constantly  exerting  a 
vigorous  action  on  each  other,  and  bringing  abont  changes  the 
most  important. 

The  earth  supports  and  nourishes  both  the  vegetable  and  animal 
world,  and  the  researches  of  Geology  have  demonstrated,  that 
there  was  once  a  time  when  neither  phints  nor  animals  existed  on 
its  surface,  and  that  prior  to  the  creation  of  either  of  these  orders, 
great  changes  nnist  have  taken  place  in  its  constitution.  As  llie 
earth  existed  thus  anterior  to  the  organized  beings  upon  it,  we 
may  infer  that  the  other  heavenly  bodies,  in  like  manner,  were 
called  into  being  before  any  of  the  organized  bodies  wliich  pro- 
bably exist  upon  them.  Heasoning,  then,  by  analogy  from  our 
earth,  we  mav  venture  to  regard  the  heavenlv  bodies  as  the  i)ri- 
mary  organized  forms,  on  whose  surface  both  aninnils  and  vege- 
*ables  find  a  })lace  and  support. 

Naixral  Philomplnj^  or  Physics,  treats  of  tl;:>  general  prnj)o> 
ties  of  vnorgeinisedhoiWes^  of  the  influences  wjiicii  Jict  upon  them, 
the  laws  they  obey,  and  of  the  external  changes  Avhicli  these 
bodies  undergo  without  affecting  their  internal  constitution. 

Chemistry,  on  the  contrary,  treats  of  the  individual  properties 


INTRODUCTION.  Hi 

of  bodies,  b^  which,  as  regards  their  constitution,  the}^  may  be 
distinguislied  one  from  another  ;  it  also  investigates  the  transfor- 
mations which  take  phice  in  the  interior  of  a  body — transforma- 
tions by  which  the  substance  of  the  body  is  altered  and  remodeled; 
and  lastly,  it  detects  and  classifies  the  laws  by  which  chemical 
changes  are  regulated. 

Natural  History^  is  that  branch  of  physical  science  which 
treats  of  organized  bodies ;  it  comprises  three  divisions,  the  one 
mechanical — the  anatomy  and  dissection  of  plants  and  animals ; 
the  second,  chemical — animal  and  vegetable  chemistry  ;  and  the 
th i  rd ,  explanatory — physiology. 

Astronomy  teaches  the  knowledge  of  the  celestial  bodies.  It  is 
divided  into  Spherical  and  Physical  astronomy.  The  former 
treats  of  the  appearances,  magnitudes,  distances,  arrangements, 
and  motions  of  the  heavenly  bodies ;  the  latter,  of  their  consti- 
tution and  physical  condition,  their  mutual  influences  and  actions 
on  each  other,  and  generally,  seeks  to  explain  tlie  causes  of  the 
celestial  phenomena. 

Again,  one  most  important  use  of  natural  science,  is  the  appH- 
cation  of  its  laws  either  to  technical  pui-jioses — mechanics^  tcrh- 
nical  chemi.sfry^  pharmacy^  <&c. ;  to  the  phenomena  of  the 
heavenly  bodies — pJiysical  astronomy;  or  to  the  various  objects 
which  pre:-ent  themselves  to  our  notice  at  or  near  the  surface  ot 
tlic  Qiivt\\—2?hysical  geography^  meteorology — and  we  may  add 
geology  also,  a  science  which  has  for  its  object  to  unfold  th*o 
history  of  our  planet  from  its  formation  to  the  present  time. 

Natural  philos<:»phy  is  a  science  of  ohservation  and  experiment, 
for  by  these  two  modes  we  deduce  the  varied  information  we 
have  acquired  about  bodies ;  by  the  former  we  notice  any 
changes  that  transpire  in  the  condition  or  relations  of  any  body 
as  they  spontaneously  arise  without  interference  on  our  part ; 
whereas,  in  the  performance  of   an    exj^eriment,   we  purposely 


14  ELEMENTS     OF     ANALYTICAL     MECHANICS. 

alter  the  natural  arrangement  of  things  to  bring  about  some  par- 
ticular condition  that  we  desire.  To  acconiplisli  this,  we  ujake 
use  of  appliances  called  j)hilosophical  or  chemical  ajjjxirutus^  the 
proper  use  and  application  of  which,  it  is  the  otiice  of  Experi- 
mental Pliysics  to  teach. 

If  we  notice  that  in  winter  water  becomes  converted  into  ice, 
we  are  said  to  make  an  observation  ;  if,  by  means  of  freezing 
mixtures  or  evaporation,  we  cause  water  to  freeze,  we  are  then 
said  to  perform  an  experiment. 

These  experiments  are  next  subjected  to  calculation,  by  whicli 
are  deduced  what  are  sometimes  called  the  laws  of  nature,  or  the 
rules  that  lihe  causes  icill  hwariahly  2^>'0(luce  like  results.  To 
express  these  laws  with  the  greatest  possible  brevity,  mathematical 
symbols  are  used.  When  it  is  not  practicable  to  represent  them 
with  mathematical  precision,  we  must  be  contented  with  infer- 
ences and  assumptions  based  on  analogies,  or  witli  probable 
explanations  or  hypotheses. 

A  hypothesis  gains  in  j)robability  the  more  neai'ly  it  accords 
with  the  ordinary  course  of  nature,  the  more  numerous  the 
experiments  on  which  it  is  founded,  and  the  more  simple  the 
explanation  it  olfers  of  the  phenomena  for  vs'hich  it  is  intended  to 
account. 

PHYSICS  OF  PONDERABLE  BODIES. 

§  1. — T\\(i  j^hysical pro].>C7'ties  of  bodies  arc  those  external  signs 
by  which  their  existence  is  made  evident  to  our  minds;  the  senses 
constitute  the  medium  through  which  this  knowledge  is  coni- 
municnted. 

All  our  senses,  however,  are  not  equally  made  use  of  for  tliis 
pni'pose;  we  are  generally  guided  in  our  decisions  b}^  the  evidence 
of  sight  and  touch.  Still  sight  alone  is  frequently  incompetent, 
as  tliere  are  ])odies  which  cannot  be  perceived  by  that  sense,  as, 
for  exanq)le,  all  colorless  gases;  again,  some  of  the  objecrs  of 
sight  ar(;   not   substantial,  as,  the  shadow,  the  image   in  a  mirror, 


INTRODUCTION  15 

spectra  formed  by  the  refraction  of  the  rays  of  light,  Sec. 
Touch,  on  the  contrary,  decides  indubitably  as  to  the  existence 
of  any  body. 

The  jDroperties  of  bodies  may  be  divided  mio prhnary  or  j9r?«- 
cijjol,  and  secondary  or  accessory.  The  former,  are  sucli  as  Ave 
find  common  to  all  bodies,  and  without  which  we  cannot  conceive 
of  their  existing ;  the  latter,  are  not  absolutely  necessary  to  our 
conception  of  a  body's  existence,  but  become  known  to  us  by 
investigation  and  experience. 

PEIMAKY   PROPERTIES. 

§  2. — ^The  primary  properties  of  all  bodies  are  extension  and 
im/penetTobility. 

Extension  is  that  property  in  consequence  of  which  every  body 
occupies  a  certain  limited  space.  It  is  the  condition  of  the 
mathematical  idea  of  a  body ;  by  it,  the  volume  or  size  of  the 
Occupied  space,  as  well  as  its  boundary,  or  Jigure,  is  deteruiined. 
The  extension  of  bodies  is  expressed  by  three  dimensions,  length, 
breadth,  and  thickness.  The  computations  from  these- data,  follow 
geometrical  rules. 

Impenetrability  is  evinced  in  the  fact,  that  one  body  cannot 
enter  into  the  space  occupied  by  another,  without  previously 
thrusting  the  latter  from  its  place. 

A  body  then,  is  whatever  occupies  space,  and  j^ossesses  exten- 
sion and  impenetrability.  One  might  be  led  to  imagine  that  tlie 
property  of  impenetrability  belonged  only  to  solids,  since  we  see 
them  penetrating  both  air  and  water  ;  but  on  closer  observation 
it  will  be  apparent  that  this  property  is  common  to  all  bodies  of 
whatever  nature.  If  a  hollow  cylinder  into  which  a  piston  fits 
accurately,  be  filled  with  water,  the  piston  cannot  be  thrust  into 
the  water,  thus  showing  it  to  be  impenetrable.  Invert  a  glass 
tuml)ler  in  any  liquid,  tlie  air,  unable  to  escape,  will  prevent  the 
liquid  h-om  occupvijig  its  ])l;!ce.  tlius  proving  the  impenetrability 


16 


ELEMENTS     OF     ANALYTICAL     MECHANICS. 


of    air.      The   diving-bell   affords  a   familiar  illustration  of    tlris 
proj^erty. 

The  difficulty  of  pouring  liquid  into  a  vessel  having  only  one 
small  hole,  arises  from  the  impenetrability  of  the  air,  as  the 
liquid  can  run  into  the  vessel  only  as  the  air  makes  its  escape. 
The  following  experiment  will  illustrate  this  fact : 

In  one  mouth  of  a  two- 
necked  bottle  insert  a  funnel 
«,  and  ic  the  other  a  siphon  h 
the  longer  leg  of  which  is  im- 
mersed in  a  glass  of  water. 
Now  lot  water  be  poured  into 
the  funnel  «,  and  it  will  be 
seen  that  in  proportion  as  this 
water  descends  into  the  vessel 
F,  the  air  makes  its  escape 
through  the  tube  5,  as  is 
proved  by  the  ascent  of  the 
bubbles  in  the  water  of  the 
tumbler. 


SECONDARY    PROPERTIES. 


The  secondary  properties  of  bodies  are  compressihility,  expa/nst- 
hility,  porosity ^  divisihility,  and  elasticity. 


§  3. — Comprcssiljility  is  that  property  of  bodies  by  virtue  of 
which  they  may  be  made  to  occupy  a  smaller  space  :  and  exjjansi- 
hilify  is  that  in  consequence  of  which  they  may  be  mjulo  to  till  a 
larger,  without  in  cither  case  altering  the  quantity  of  matter  they 
contain. 

Both  changes  are  produced  in  all  bodies,  as  we  shall  presently 
i?ee,  by  change  of  temperature  ;  many  bodies  may  also  be  reduced 
m  bulk  by  pressure,  percussion,  &c. 


INTRODUCTION.  17 

§  4-. — Since  all  bodies  admit  of  compression  and  expansion,  it 
follows  of  necessity,  tluit  there  must  be  interstices  between  their 
minutest  particles ;  and  that  property  of  a  body  by  which  its 
constituent  elements  do  not  completely  fill  the  space  witluii  its 
exterior  boundary,  but  leaves  holes  or  pores  between  them,  is 
called  jjorosity.  The  pores  of  one  body  are  often  tilled  with  some 
other  body,  and  the  pores  of  this  with  a  third,  as  in  the  case  of  a 
sponge  containing  water,  and  the  water,  in  its  turn,  containing 
air,  and  so  on  till  we  come  to  the  most  subtle  of  substances, 
ether,  which  is  supposed  to  pervade  all  bodies  and  all  space. 

In  many  cases  the  pores  are  visible  to  the  naked  eye  ;  ia  others 
they  are  only  seen  by  the  aid  of  the  microscojie,  and  when  so 
minute  as  to  elude  the  power  of  this  instrument,  their  existence 
may  be  inferred  from  experiment.  Sponge,  cork,  wood,  bread, 
&c.,  are  bodies  whose  pores  are  noticed  by  the  naked  eye.'  The 
human  skin  appears  full  of  them,  when  viewed  with  the  magni- 
fying glass ;  the  porosity  of  water  is  shown  by  the  ascent  of  air 
bubbles  w^hen  the  temperature  is  raised. 

§  5. — The  divisibility  of  bodies  is  that  property  in  consequence 
of  which,  by  various  mechanical  means,  such  as  beating,  pound- 
ing, grinding,  &c.,  we  can  reduce  them  to  particles  homogeneous 
to  each  other,  and  to  the  entire  mass ;  and  these  again  to  smaller, 
and  so  on. 

B}"  the  aid  of  matnematical  processes,  the  mind  may  be  led  to 
admit  the  infinite  divisibility  of  bodies,  though  their  practical 
division,  by  mechanical  means,  is  subject  to  limitation.  Many 
examples,  however,  prove  that  it  may  be  carried  to  an  incredible 
extent.  We  are  furnished  with  numerous  instances  among  nat- 
ural objects,  whose  existence  can  only  be  detected  by  means  of 
the  most  acute  senses,  assisted  by  the  most  powerful  artificial 
aids ;  the  size  of  such  objects  can  only  be  calculated  approxi- 
mately. 

Mechanical  subdivisions  for  purposes  connected  with  the  arts 

are  exemplified  in  the  grinding  of  corn,  the  pulverizing  of  sul- 

2 


18      ELEMENTS  OF  ANALYTICAL  MECHANICS. 

pliiir,  charcoal,  and  saltpetre,  for  the  maimfactiire  of  gunpcwder ; 
and  Homeopathy  affords  a  remarkable  instance  of  the  extended 
application  of  this  property  of  bodies. 

Some  metals,  particularly  gold  and  silver,  are  susceptible  of  a 
very  great  divisibility.  In  the  common  gold  lace,  the  silver 
thread  of  which  it  is  composed  is  covered  with  gold  so  attenuated, 
that  the  quantity  contained  in  a  foot  of  the  thread  weighs  less 
than  60^0  of  a  grain.  An  inch  of  such  thread  will  therefore 
contain  yaffoo  ^^  ^  grain  of  gold;  and  if  the  inch  be  divided  into 
100  equal  parts,  each  of  which  would  be  distinctly  visible  to  the 
eye,  the  quantity  of  the  precious  metal  in  each  of  such  pieces 
would  be  "720^000  of  a  grain.  One  of  these  particles  examined 
through  a  microscope  of  500  times  magnifyiug  power,  will  appear 
500  times  as  long,  and  the  gold  covering  it  will  be  visible,  having 
been  divided  into  3,600,000,000  parts,  each  of  which  exhibits  all 
the  characteristics  of  this  metal,  its  color,  density,  &c. 

Dyes  are  likewise  susceptible  of  an  incredible  divisibility. 
^yith  1  grain  of  blue  carmine,  10  lbs.  of  water  may  be  tiuged 
blue.  These  10  lbs.  of  water  contain  about  617,000  drops.  Sup- 
posing now,  that  100  particles  of  carmine  are  required  in  each 
drop  to  produce  a  uniform  tint,  it  follows  that  this  one  grain  of 
carmine  has  been  subdivided  62  millions  of  times. 

According  to  Biot,  the  thread  by  which  a  spider  lets  herself 
down  is  composed  of  more  than  5000  single  threads.  The  single 
threads  of  the  silkworm  are  also  of  an  extreme  fineness. 

Om*  blood,  which  appears  like  a  uniform  red  mass,  consists  of 
small  red  globules  swimming  in  a  transparent  fluid  called  serum. 
The  diameter  of  one  of  these  globules  does  not  exceed  the  4000th 
part  of  an  inch  :  whence  it  follows  that  one  drop  of  blood,  such 
as  would  hang  from  the  point  of  a  needle,  contain.s  at  least  one 
million  of  these  globules. 

But  more  surprising  than  all,  is  the  microcosm  (.n'organizet!  nature 
in  the  Infusoria,  for  more  exact  acquaintance  with  which  we  are 
indebted  to  the  unwearied  researches  of  Ehrenbers:.    Of  these  crea- 


-^^^  ^i 


INTRODUCTION-.  19 

tures.  whicli  for  tlie  most  part  we  can  see  only  \)\  the  aid  of  the 
microscope,  there  exist  many  species  so  small  that  millions  piled  on 
each  other  would  not  eqnal  a  single  grain  of  sand,  and  thousands 
might  swim  at  once  through  the  eye  of  the  finest  needle.  Tlie 
coats-of-mail  and  shells  of  these  animalcules  exist  in  such  prodi- 
gious quantities  on  our  earth  that,  according  to  Ehrenberg's  inves- 
tigations, pretty  extensive  strata  of  rocks,  as,  for  instance,  the 
smooth  slate  near  Bilin,  in  Bohemia,  consist  almost  entirely  of 
them.  By  microscopic  measurements  1  cubic  line  of  this  slate  con- 
tains about  23  millions,  and  1  cubic  inch  about  41,000  millions  of 
these  animals.  As  a  cubic  inch  of  this  slate  weic-hs  220  o-rains, 
187  millions  of  these  shells  niust  go  to  a  grain,  each  of  which 
would  consequently  weigh  about  the  j-jy  millionth  part  of  a  grain. 
Conceive  further  that  each  of  these  animalcules,  as  microscopic 
investigations  have  proved,  has  his  limbs,  entrails,  &c.,  the  possi- 
bility vanishes  of  our  forming  the  most  remote  conception  of  the 
dimensions  of  these  organic  forms. 

In  cases  where  our  finest  instruments  are  unable  to  render  us 
the  least  aid  in  estimating  the  minuteness  of  bodies,  or  the 
degree  of  subdivision  attained;  in  other  words,  when  bodies 
evade  the  perception  of  our  sight  and  touch,  our  olfactory  nerves 
frequently  detect  the  presence  of  matter  in  tlie  atmosphere,  of 
which  no  chemical  analysis  could  afiibrd  us  the  slightest  inti 
mation. 

Thus,  for  instance,  a  single  grain  of  musk  diffuses  in  a  large 
and  airy  room  a  powerful  scent  that  frequently  lasts  for  years ; 
and  papers  laid  near  musk  will  make  a  voyage  to  the  East  Indies  • 
and  back  without  losing  the  smell.  Imagine  now,  how  many  par- 
ticles of  musk  must  radiate  from  such  a  body  every  second,  in 
order  to  render  the  scent  pcrceiDtible  in  all  directions,  and  you 
will  be  astonished  at  their  number  and  minuteness. 

In  like  manner  a  single  drop  of  oil  of  lavender,  evaporated  in  a 
spoon  over  a  spirit-lamp,  fills  a  large  room  with  its  fragrance  for 
a  length  of  time. 


20  ELEMENTS     OF     ANALYTICAL    MECHANICS. 

§  6. — Elasticity  is  the  name  given  to  that  property  of  bodies, 
by  virtue  of  Avhich  they  resume  of  themselves  their  figure  and 
dimensions,  when  these  have  been  changed  or  altered  by  any 
extraneous  cause.  Different  bodies  possess  this  property  in  very 
different  degrees,  and  retain  it  with  very  unequal  tenacity. 

The  following  are  a  few  out  of  a  large  number  of  higlily 
elastic  solid  bodies;  viz.,  glass,  tempered  steel,  ivory,  whale 
bone,  itc. 

Let  an  ivory  ball  fall  on  a  marble  slab  smeared  Vvith  some  col- 
oring matter.  The  point  struck  by  the  ball  shows  a  round  speck 
which  will  have  imprinted  itself  on  the  surface  of  the  ivory  with 
out  its  spherical  form  being  at  all  impaired. 

Fluids  under  peculiar  circumstances  exhibit  considerable  elas- 
ticity ;  this  is  particularly  the  case  with  melted  metals,  more 
evidently  sometimes  than  in  their  solid  state.  The  following 
experiment  illustrates  this   fact   with   regard   to   antimony  and 

bismuth. 

Place  a  little  antimony  and  bismuth  on  a  piece  of  charcoal,  so 
that  the  mass  when  melted  shall  be  about  the  size  of  a  pepper- 
corn ;  raise  it  by  means  of  a  blowpipe  to  a  white  heat,  and  then 
turn  the  ball  on  a  sheet  of  paper  so  folded  as  to  have  a  raised 
edge  all  round.  As  soon  as  the  liquid  metal  falls,  it  divides  itself 
into  many  minute  globules,  Avhich  hop  about  upon  the  paper  and 
continue  visible  for  some  time,  as  they  cool  but  slowly ;  the  j^oints 
at  which  they  strike  the  paper,  and  their  course  upon  it,  will  be 
marked  by  black  dots  and  lines. 

The  recoil  of  cannon-balls  is  owing  to  the  elasticity  of  the  iron 
and  that  of  the  bodies  struck  by  them. 

'  FORCE. 

§  7. — Whatever  t«idft-4o  change^the  actual  state  of  a  body,  in 
respect  to  rest  or  motion,  is  called  a  force.  If  a  body,  for 
instance,  be  at  rest,  the  influence  which  changes  or  tends  to 
change  this  state  to  that  of  motion,  is  called  force.     Again,  if  a 


INTRODUCTION.  21 

body  be  already  in  :notion,  auj  cause  which   urges  it  to  luovo 
faster  or  slower,  is  caMod/oiVc. 

Of  the  actual  nature  of  forces  we  are  ignorant;  we  know  of 
tiieir  existence  only  by  the  etfects  they  produce,  and  with  these 
we  become  acquainted  solely  through  tlie  medium  of  the  sense:;. 
Hence,  while  their  operations  are  going  on,  they  appear  to  us 
always  in  connection  wdtli  some  body  which,  in  some  way  or 
otlier,  affects  our  senses. 

§  8.— "We  shall  find,  though  not  always  upon  superficial  inspec- 
tion, that  the  approaching  and  receding  of  bodies  or  of  their  com- 
ponent parts,  Avhen  this  takes  place  apparently  of  their  own 
accord,  are  but  the  results  produced  by  the  various  forces  that 
come  under  our  notice.  In  other  words,  that  the  nuivcrsally  ope- 
rating forces  are  those  of  attraction  and  of  repulsion. 

§  9.— Expei'ience  proves  that  these  nniversal  forces  are  at  work 
in  two.  essentially  different  modes.  They  are  operating  either  in 
the  interior  of  a  body,  amidst  the  elements  which  compose  it,  or 
they  extend  their  intluence  through  a  wide  ]'ange,  and  act  upon 
bodies  in  the  aggregate ;  the  former  distinguished  as  Atomical 
or  Molecular  action,  the  latter  as  the  Attraction  of  gravitation. 

§  10. — Molecular  forces  and  the  force  of  gravitation,  often  co 
exist,  and  qualify  each  other's  action,  giving  rise  to  those  attrac- 
tions and  repulsions  of  bodies  exhibited  at  their  surfaces  vxdien 
brought  into  sensible  contact.  This  resultant  action  is  calk'd  tlie 
force  of  cohesion  or  of  dissolution,  according  as  it  tends  to  unite 
different  bodies,  oi-  the  elements  of  the  same  body,  more  closely, 
or  to  sc})arate  them  more  \videly. 

§  11. — Inertia  is  that  principle  by  which  a  body  resists  all 
change  of  its  condition,  in  respect  to  rest  or  motion.  If  a  body  be 
at  rest,  it  will,  in  the  act  of  yielding  its  condition  of  rest,  while 
under  the  action  of  any  force,  o])pose  a  resistance ;  so  also,  it  a 
body  be  in  motion,  and  be  urged  to  move  faster  or  slower,  it  will, 


22  ELEMENTS     OF     ANALYTICAL     MECHANICS. 

duj'ing  the  act  of  clianging,  oppose  an  equal  resistance  for  everj 
equal  amount  of  change.  AYe  derive  our  knowledge  of  this  prin- 
ciple solely  from  experience  ;  it  is  found  to  be  common  to  all 
bodies;  it  is  in  its  nature  conservative,  though  passive  in  charac- 
ter, being  only  exerted  to  preserve  the  state  of  rest  or  of  particu 
lar  motion  which  a  body  has,  by  resisting  all  variation  therein. 
'Whenever  any  force  acts  upon  a  free  body,  the  inertia  of  the 
latter  reacts,  and  tliis  action  and  reaction  are  equal  and  contrary. 

§  12. — Molecular  action  chiefly  determines  the  forms  of  bodies. 
All  bodies  are  ]-egarded  as  collections  or  aggregates  of  minute  ele- 
ments, called  atoms,  and  are  formed  by  the  attractive  and  repul- 
sive forces  acting  upon  them  at  immeasurably  small  distances. 

Several  hypotheses  have  been  proposed  to  explain  the  constitu- 
tion of  a  body,  and  the  mode  of  its  formation.  The  most  remark- 
able of  these  was  by  Boscovich,  about  the  middle  of  the  last  cen- 
tury. Its  great  fertility  in  the  explanations  it  aifords  of  tlie  prop- 
erties of  what  is  called  tangible  matter,  and  its  harmouy  with  the 
laws  of  motion,  entitle  it  to  a  much  larger  space  than  can  be 
found  for  it  in  a  work  lilce  this.  Enougli  may  be  stated,  however, 
to  enable  the  attentive  reader  to  seize  its  leading  features,  and  to 
appreciate  its  competency  to  explain  the  phenomena  of  nature. 

1.  All  matter  consists  of  indivisible  and  inextended  atoms. 

2.  These  atoms  are  endowed  with  attractive  and  repulsive 
forces,  varying  both  in  intensity  and  direction  by  a  change  of  dis- 
tance, so  that  at  one  distance  two  atoms  attract  each  other,  and  at 
another  distance  they  repel. 

3.  This  law  of  variation  is  the  same  in  all  atoms.  It  is,  there- 
fore, mutual :  for  the  distance  of  atom  a  from  atom  h,  being  the 
same  as  that  of  h  from  a,  if  a  attract  l>,  h  must  attract  a  with 
precisely  the  same  force. 

4.  At  all  considerable  or  sensible  distances,  these  mutual  forces 
are  attractive  and  sensibly  proportional  to  the  square  of  the  dis- 
tance inversely.     It  is  the  attraction  called  (jramtation. 

5    In  the  Muall  and  insensible  distances  in  which  sensible  con- 


INTRODUCTION.  23 

tact  is  observed,  and  which  do  not  exceed  the  1000th  or  1500th 
part  of  an  inch,  there  are  many  alternations  of  attraction  and 
repulsion,  according  as  the  distance  of  the  atoms  is  chano-ed. 
Consequently,  there  are  many  situations  within  this  narrow  limit, 
in  which  two  atoms  neither  attract  nor  re^^el. 

6.  The  force  which  is  exerted  between  two  atoms  when  their 
distance  is  diminished  without  end,  and  is  just  vanishing,  is  an 
insuperable  repulsion,  so  that  no  force  whatever  can  press  two 
atoms  into  mathematical  contact. 

Such,  according  to  Boscovich,  is  the  constitution  of  a  material 
atom  and  the  whole  of  its  constitution,  and  the  immediate  efficient 
cause  of  all  its  properties. 

Two  or  more  atoms  may  be  so  situated,  in  respect  to  position 
and  distance,  as  to  constitute  a  molecule.  Two  or  more  molecules 
may  constitute  ixparticle.     The  particles  constitute  a  hody. 

Now,  if  to  these  centres,  or  loci  of  the  qualities  of  what  is 
termed  matter,  we  attribute  the  property  called  inertia,  we  have 
all  the  conditions  requisite  to  explain,  or  arrange  in  the  order  of 
antecedent  and  consequent,  the  various  operations  of  the  physical 
world. 

Boscovich  represents  his  law  of  atomical  action  by  what  may 
be  called  an  exponential  curve.     Let  the  distance  of  two  atoms 


c'' 


Cr  <J 


be  estimated  on  the  line  G  A  O,  A  being  the  situation  of  one  ol 
them,  while  the  other  is  placed  anywhere  on  this  line.  When 
placed  at  ?',  for  example,  we  may  suppose  that  it  is  attracted  by 
A,  with  a  cei'tain  intensity.     We  can  represent  this  intensity  by 


24  ELEMENTS     OF    ANALYTICAL    MECHANICS. 

the  length  of  the  line  i  l^  perpendicular  to  A  C^  and  caii  express 
the  direction  of  the  force,  namely,  from  i  to  A^  because  it  is 
attractive,  by  placing  i  I  above  the  axis  A  C.  Should  the  atom 
be  at  m,  and  be  repelled  by  A,  we  can  exjjress  the  intensity  of 
repulsion  by  on  n^  and  its  direction  from  m  towards  G  by  placing 
m  n  below  the  axis. 

This  may  be  supposed  for  every  point  on  the  axis,  and  a  curve 
drawn  through  the  extremities  of  all  the  perpendicular  ordinates. 
This  will  be  the  exponential  curve  or  scale  of  force. 

As  there  are  supposed  a  great  many  alternations  of  attractions 
and  repulsions,  the  cur\e  must  consist  of  many  branches  lying  on 
opposite  sides  of  the  axis,  and  must  therefore  cross  it  at  6",  D', 
6"',  D'\  &c.,  and  at  G.  All  these  m-e  supposed  to  be  contained 
within  a  very  small  fraction  of  an  inch. 

Beyond  this  distance,  which  terminates  at  G,  the  force  is 
always  attractive,  and  is  called  the  force  of  gravitation,  the  maxi- 
mum intensity  of  which  occurs  at  g,  and  is  expressed  by  the 
length  of  the  ordinate  G'g.  Further  on,  the  ordinates  are  sensibly 
proportional  to  the  square  of  their  distances  from  A,  inversely. 
The  branch  G'  G"  has  the  line  A  C,  therefore,  for  its  asymptote. 

"Within  the  limit  A  O  there  is  repulsion,  which  becomes  infi- 
nite, wlien  the  distance  from  A  is  zero ;  whence  the  branch  O  D" 
has  the  perpendicular  axis,  A  y,  for  its  asymptote. 

An  atom  being  placed  at  G,  and  then  disturbed  so  as  to  move 
it  in  the  direction  towards  A,  will  be  repelled,  the  ordinates  of  the 
curve  being  below  the  axis ;  if  disturbed  so  as  to  move  it  from 
A,  it  will  be  attracted,  the  corresponding  ordinates  being  above 
the  axis.  The  point  G  is  therefore  a  position  in  which  the  atom 
is  neither  attracted  nor  repelled,  and  to  which  it  will  tend  to 
return  Avhen  slightly  removed  in  either  direction,  and  is  called  the 
Urait  of  gravitatiooi. 

If  tiie  atom  be  at  C,  or  C",  &c.,  and  be  moved  ever  so  little 
towards  A,  it  will  be  repelled,  and  when  the  disturl)ing  cause  is 
removed,  will  fly  back;  if  moved  from  A,  it  will  be  attracted 


liS'TROD  UCTION. 


25 


and  return.  Hence  6",  C",  &c.,  arc  positions  similar  to  G^  and  <ivg 
called  limits  of  cohesion,  C  being  termed  the  last  limit  of  coke' 
sion.  An  atom  situated  at  any  one  of  these  points  will,  with  that 
at  A,  constitute  £i  jyermanent  molecule  of  the  siuiplest  kind. 

On  the  contrary,  if  an  atov.i  he  placed  at  D\  oj-  D'\  tVc,  a;id 
be  then  slightly  disturbed  in  the  direction  either  from  or  Towards 
J-,  the  action  of  the  atom  at  A  will  cause  it  to  recede  still  further 
frv>m  its  first  position,  till  it  reaches  a  limit  of  cohesion.  The 
points  D ,  D'\  tfcc.j  are  also  positions  of  indilference,  in  which  the 
atom  will  be  neither  attracted  nor  repeh'^xl  by  that  at  A,  but  tliey 
ditt'er  from  G,  G  6"',  &c.,  in  this,  that  an  atom  being  ever  :;o  little 
removed  from  one  of  them  has  no  dis]30sition  to  return  to  it 
again ;  these  points  are  called  limits  of  dissolution.  An  atom 
situated  in  one  of  them  cannot,  therefore,  constitute,  with  that  at 
A,  a  permanent  molecule,  but  the  slightest  disturbance  v.-ill  de- 
stroy it. 

It  is  easy  to  infer,  from  what  has  been  said,  how  three,  four, 
^tc,  atoms  may  combine  to  form  molecules  of  different  orders  of 
complexity,  aud  how  these  again  may  be  arranged  so  as  1)}'  their 
action  upon  each  other  to  form  particles.  Our  limits  will  imt 
permit  us  to  dwell  ujDon  these  points,  but  we  cannot  dismiss  the 
subject  without  suggesting  one  of  its  most  interesting  conse- 
quences. 

According  to  the  highest  authority  on  the  subject,  the  sun  and 
other  heavenly  bodies  have  been  formed  by  the  gradual  subsi- 
dence of  a  vast  nebula  towards  its  centre.     Its  molecules  fo^'ced 


26  ELEMENTS     OF    ANALYTICAL    MECHANICS. 

by  tlieir  gravitating  action  within  their  neutral  limits,  are  in  a 
state  of  tension,  which  is  the  more  intense  as  the  accumulation  is 
greater ;  and  the  molecular  agitations  in  the  sun  caused  by  the 
successive  depositions  at  its  surface,  make  this  body,  in  conse- 
quence of  its  vast  size,  the  principal  and  perpetual  fountain  of 
that  incessant  stream  of  ethereal  waves  which  are  now  generally 
believed  to  constitute  the  essence  of  light  and  heat.  The  same 
principle  furnishes  an  explanation  of  the  internal  heat  of  our 
earth  which,  together  with  all  the  heavenly  bodies,  would  doubt- 
less appear  self-luminous  were  the  acuteness  of  our  sense  of  sight 
increased  beyond  its  present  limit  in  the  same  proportion  that  the 
sun  exceeds  the  largest  of  these  bodies.  The  sun  far  transcends 
all  the  other  bodies  of  our  system  in  regard  to  heat  and  light,  and 
is  in  a  state  of  incandesGence  simply  because  of  his  vastly  greater 
size. 

§  13. — ^The  molecular  forces  are  the  effective  causes  which 
hold  together  the  particles  of  bodies.  Through  them,  the 
molecules  approach  to  a  certain  distance  wdiere  they  gain  a 
position  of  rest  with  respect  to  each  other.  The  j)ower  with 
which  the  particles  adhere  in  these  relative  positions,  is  called, 
as  we  have  seen,  cohesion.  This  force  is  measured  by  the 
resistance  it  offers  to  mechanical  separation  of  the  parts  of 
bodies  from  each  other. 

The  different  states  of  matter  result  from  certain  definite 
relations  under  which  the  molecular  attractions  and  repulsions 
establish  their  equilibrium  ;  there  are  three  cases,  viz.,  two 
extremes  and  one  mean.  The  first  extreme  is  that  in  which 
attraction  predominates  among  the  atoms  ;  this  produces  the 
solid  state.  In  the  other  extreme  repulsion  prevails,  and  the 
gaseous  form  is  the  consequence.  The  mean  obtains  when 
neither  of  these  forces  is  in  excess,  and  then  matter  presents 
itself  under  tlie  liquid  form. 

Let  A   represent   the   attraction   and  li   the   repulsion,   then 


INTRODUCTION.  27 

the  three  aggregate  forms  may  be  expressed  by  the  followinc^ 
formulae : 

A>I^  solid, 

A<  n  gas, 

A  —  J2  liquid. 

These  three  forms  or  conditions  of  matter  may,  for  the  mos 
part,  be  readily  distinguished  by  certain  extei-nal  peculiarities; 
there  are,  however,  especially  between  solids  and  liquids,  so 
many  imperceptible  degrees  of  approximation,  that  it  is  some- 
times difficult  to  decide  where  the  one  form  ends  and  the 
other  begins.  It  is  further  an  ascertained  fact  that  many 
bodies,  (j^erhaps  all,)  as  for  instance,  water,  are  capable  of 
assuming  all  three  forms  of  aggregation. 

Thus,  supposing  that  the  relative  intensity  of  the  molecular 
forces  determines  these  three  forms  of  matter,  it  follows  from 
what  has  been  said  above,  that  this  term  may  vary  in  the 
same  body. 

The  peculiar  properties  belonging  to  each  of  these  states 
will  be  explained  when  solid,  liquid,  and  aeriform  bodies  come 
severally  under  our  notice, 

§  14. — The  molecular  forces  may  so  act  upon  the  atoms  of 
dissimilar  bodies  as  to  cause  a  new  combination  or  union  of 
their  atoms.  This  may  also  produce  a  separation  between  the 
combined  atoms  or  molecules  in  such  manner  as  to  entirely 
change  the  individual  properties  of  the  bodies.  Such  efforts 
of  the  molecular  forces  are  called  cheinical  action^  and  the 
disi30sition  to  exert  these  efforts,  on  account  of  the  peculiar 
state  of  aggregations  of  the  ultimate  atoms  of  different  bodies, 
■jhemical  affinity.  > 

§  15. — Beyond  the  last  limit  of  gravitation,  atoms  attract 
each  other :  hence,  all  the  atoms  of  one  body  attract  each 
atom  of   another,  and  vice  versa :    thus   gi\  ing   rise    to    attrac- 


28  ELEMENTS     OF     ANALYTICAL    MECHANICS. 

cions  between  bodies  of  sensible  magnitudes  through  sensible 
distances.  The  intensities  of  tliese  attractions  are  proportional 
to  the  number  of  atoms  in  the  attracting  body  directly,  and 
to  the  square  of  the  distance  between  the  bodies  inversely, 

1 16. — ^The  term  universal  gravitation  is  applied  to  this  foi'ce 
■when  it  is  intended  to  express  the  action  of  the  heavenly 
bodies  on  each  other  ;  and  that  of  terrestrial  gravitation  or 
simply  gravity^  wliere  we  wish  to  express  tlie  action  of  the 
earth  upon  the  bodies  forming  with  itself  one  whole.  The 
force  is  ahvays  of  the  same  kind  however,  and  varies  in 
intensity  only  by  reason  of  a  difference  in  the  number  of 
atoms  and  their  distances.  Its  effect  is  always  to  generate 
motion  when   the  bodies  are   free   to  move. 

Gravity,  then,  is  a  property  common  to  all  terrestrial  bodies, 
since  they  constantly  exhibit  a  tendency  to  approach  the 
earth  and  its  centre.  In  consequence  of  this  tendency,  all 
bodies,  unless  supported,  fall  to  the  surface  of  the  earth,  and 
if  prevented  by  any  other  bodies  from  doing  so,  they  exert  a 
pressure  on  these  latter. 

This  is  one  of  the  most  important  properties  of  terrestrial 
bodies,  and  the  cause  of  many  phenomena,  of  which  a  fulle: 
explanation  will  be  given  hereafter. 

§17. — The  mass  of  a  body  is  the  number  of  atoms  it  con- 
tains, as  compared  with  the  number  contained  in  a  unit  of 
volume  of  some  standard  substance  assumed  as  nnity.  Tlie 
unit  T)f  volume  is  usually  a  cubic  foot,  and  the  standard  sub- 
stance is  distilled  water  at  the  temperature  of  3S°,Yo  Falnvii- 
lieit.  Hence,  the  number  of  atoms  contained  in  a  cubic  font 
of  distilled  water  at  38°,75  Fahrenheit,  is  the  unit  of  mass. 

The  attraction  of  the  earth  upon  tlie  atoms  of  bodies  at  its 
surface,  imparts  to  tliese  bodies,  weigJit ;    and  if  g   denote   the 


INTRODUCTION.  ^  29 


weight  of  a  nnit  of  mass,  31,  tlie  numbei*  of   imits  of  mass  in 
the  entire  body,  and  TF,  its  entire  weiglit,  then  -svill      i>J  i  id  t  i'V 

W  =  21.g (1) 

§18. — Density  is  a  term  employed  to  denote  the  degree  of 
proximity  of  the  atoms  of  a  body.  Its  measure  is  tb.e  ratio 
arising  from  dividing  the  nnml^er  of  atoms  the  body  contains, 
by  the  number  contained  in  an  equal  volume  of  some  standard 
substance  whose  density  is  assumed  as  unity.  The  standard 
substance  usually  tal^en,  is  distilled  water  at  the  temperature 
of  38°,T5  Fahrenheit.  Hence,  the  weights  of  eqnal  volumes  of 
tv.-o  bodies  being  proportional  to  the  number  of  atoms  they 
contain,  the  density  of  any  body,  as  that  of  a  piece  of  gold,  is 
found  by  dividing  its  weight  by  that  of  an  eqnal  volume  of 
distilled  water  at  38°,T5  Fahrenheit. 

Denote  the  density  of  any  body  by  D,  its  volume  by  F, 
and  its  mass  by  J/,  then  will 

M=  V.D (ly 

which  in  Equation  (1),  gives 

W  =  V.D.rj    .  (2)     ■ 

§19. — That  branch  of  science  which  treats  of  the  action  of 
forces  on  bodies,  is  called  Mechanics.  And  for  reasons  wiiich 
v.-ill  be  explained  in  the  proper  place,  this  subject  will  l)e 
treated  nnder  the  general  heads  of  Meclmnics  of  Solids^  Me- 
chanics of  Fluids,  and  Mechanics  of  Molecides. 

')  rr, :  U  \ 


L^^ 


rn/y 


c^ '-  fvco-c*z^ziniOX  - 


PAET    I. 


MECHANICS    OP    SOLIDS 


SPACE,    TIME,     MOTION,     AND     FORCE. 

§  20. — Spate  is  indefinite  extension,  without  limit,  and  contains  all 
Dodies. 

§21. — Time  is  any  limited  portion  of  duration.  We  may  conceive 
of  a  time  which  is  longer  or  shorter  than  a  given  time.  Time  has, 
therefore,   magnitude,  as  well   as  lines,  areas,  &c. 

To  measure  a  given  time,  it  is  only  necessary  to  assume  a  cert;! in 
interval  of  time  as  unity,  and  to  express,  by  a  number,  how  ol'tiMi 
this  unit  is  contained  in  the  given  time.  When  we  give  to  llii< 
number  the  particular  name  of  the  unit,  as  /;o?/?-,  minute,  second.  iScc, 
we  have  a  complete  expression  for  time. 

The  Instruments  usually  employed  in  measuring  time  are  clock'i, 
chronometers.,  and  common  watches^  which  are  too  well  known  to  need 
a  description  in  a  work  like  this. 

The  smallest  division  of  time  indicated  by. these  time-pieces  is  the 
second,  of  which  there  are  GO  in  a  minute,  3600  in  an  hour,  and 
H<)400  in  a  day  ;  and  chronometers,  which  are  nothing  more  than  a 
species  of  watch,  have  been  brought  to  such  perfection  as  not  to  vary 
in   their  rate  a  half  a  second  in   365  days,  or  31536000  seconds. 

Thus  the  number  of  hours,  minutes,  or  seconds,  between  any  two 
events   or   instants,   may   be   estimated   with    as    much   precision   and 


MECHANICS    OF    SOLIDS.  81 

ease  as  the  number  of  yards,  feet,  or  inches  between  the  extremities 
of  any  given   distance.  « 

Time  may  be  represented  by 
lines,  by.  laying  off  upon  a 
given   right  line  A  B,  the  equal 

distances  from   0   to   1,    1    to    2,  o     ^     ^     :^      ^     s      ,•      7     i 

2  to  3,  &c.,  each  one  of  these 
equal  distances  representing  the 
unit  of  time. 

A  second  is  usually  taken  as  the  unit  of  time,  and  a  foot  as  the 
linear  unit. 

§  22. — A  body  is  in  a  state  of  absolute  rest  when  it  continues  in  the 
same  place  in  space.  There  is  perhaps  no  body  absolutely  at  rest; 
our  earth  being  in  motion  about  the  sun,  nothing  connected  with  it 
can  be  at  rest.  Eest  must,  therefore,  be  considered  but  as  a  relative 
term.  A  body  is  said  to  be  at  rest,  w^hen  it  preserves  the  same 
position  in  respect  to  other  bodies  which  we  may  regard  as  fixed. 
A  body,  for  example,  which  continues  in  the  same  place  in  a  boat, 
is  said  to  be  at  rest  in  relation  to  the  boat,  although  the  boat  itself 
may  be  in  motion  in  rcljition  to  the  banks  of  a  river  on  whose  sur- 
face it  is  floating. 

§2o. — A  body  is  in  motion  when  it  occupies  successively  different 
positions  in  space.  Motion,  like  rest,  is  but  relative.  A  body  is  in 
motion  when  it  changes  its  place  in  reference  to  those  which  we 
may  regard  as  fixed. 

Motion  is  essentially  continuous ;  that  is.  a  body  cannot  pass  from 
one  position  to  another  without  passing  through  ca  series  of  interme- 
diate positions  ;  a  point,  in  motion,  therefore  describes  a  continuous 
line. 

When  we  speak  of  the  path  described  by  a  body,  we  arc  to 
understand  that  of  a  certain  point  connected  with  the  body.  Thus, 
the  path  of  a  ball,  is  that  of  its  centi'e. 

§  24. — The  motion  of  a  body  is  said  to  be  curvilinear  or  rectilinear, 
according   a's    th^    path    described  is  a  cvrve  or  riglit  line.      Motion    is 


33  ELEMENTS     OF     ANALYTICAL    MECHANICS. 

either  uniform  or  varied.  A  body  is  said  to  have  vniform  motion 
Nvhen  it  passes  ^over  equal  spaces  in  equal  successive  portions  of  time : 
and  it  is  said  to  have  varied  motion  when  it  passes  over  unequal 
spaces  in  equal  successive  portions  of  time.  The  motion  is  said  to 
be  accelerated  when  the  successive  increments  of  space  in  equal 
times  become  greater  and  greater.  It  is  retarded  when  these  incrc 
ments   become   smaller   and   smaller. 

I  25. —  Velocitij  is  the  rate  of  a  body's  motion.  Velocity  is  mea- 
sured  by   the  length  of  path  described  uniformly  in  a  unit  of  time. 

§26. — The  spaces  described  in  equal  successive  portions  of  time 
being  equal  in  uniform  motion,  it  is  plain  that  the  length  of  path 
described  in  any  time  vrill  be  equal  to  that  described  in  a  unit  of  time 
repeated  as  many  times  as  there  are  units  in  the  time.  Let  v  denote 
the  velocity,  t  the  time,  and  ^^  the  length  of  path  described,  then  Mill 

s^v.t, (3) 

If  the  position  of  the  body  be  referred  to  any  assumed  origin 
whose  distance  from  the  point  Avhere  the  motion  begins,  estimated 
in   the   direction    of  the   path   described,    be   denoted   by  S,  then   will 

s  z^  S  +  v.t (4) 

Equation  (3)  sIioavs  that  in  uniform  motion,  the  space  described 
is  always  equal  to  the  product  of  tlie  time  into  the  velocity ;  that  the 
sjyaces  described  by  different  bodies  moving  ivith  different  velocities  during 
the  same  time^  are  to  each  other  as  the  velocities;  and  that  xohen  the 
velocities  are    the   same,    the  spaces   are    to    each   other  as   the   times. 

§  27. — Differentiating   Equation    (3)    or    (4),    we   find 

ds  .^. 

■  dt='-^ ('^ 

that    is   to   say,    the   velocity  is  equal   to  the  first  differential  co-efficient 
of  the   space   regarded  as  a  function  of  the  time. 

Dividing  both  members   of  Equation   (3)   by    /,   we  have 

T  =  ^ («) 


MECHANICS     OF    SOLIDS.  33 

which  shows  that,  in  uniform  motion,  the  -velocity  is  equal  to  the  whole 
space   divided   by    the    time    in  which    it   is    described. 

§  28. — Matter  on  the  earth,  in  its  unorganized  ?tatc,  is  inanimate  or 
inert.     Tt  cannot  give  itself  motion,  nor  can  it  change  of  itseh"  the  motion 
which  it  may  have  received. 
A  body  at  rest   will   forever 
remain    so    miless    disturbed 

by   something    extraneous   to  ^ '' 

itself ;    or  if  it  be  in  motion 
in  any   direction,   as    from    a 

to  6,  it  will  continue,  after  arriving  at  6,  to  move  towards  c  in  the 
prolongation  of  ab  ;  for  having  arrived  at  b,  there  is  no  leason  Avhy 
it  should  deviate  to  one  side  more  than  another.  Moreover,  if  the 
body  have  a  certain  velocity  at  b,  it  will  retain  this  velocity  unaltered, 
since  no  reason  can  be  assigned  why  it  should  be  increased  rather 
than  diminished  in  the  absence  of  all  extraneous  causes. 

If  a  billiard-ball,  thrown  upon  the  table,  seem  to  diminish  its 
rate  of  motion  till  it  stops,  it  is  because  its  motion  is  resisted  by 
the  cloth  and  the  atmosphere.  If  a  body  thrown  vertically  down- 
ward seem  to  increase  its  velocity,  it  is  because  its  Aveight  is  inces- 
santly urging  it  onwaj'd.  If  the  direction  of  the  motion  of  a  stone, 
thrown  into  the  air,  seem  continually  to  change,  it  is  because  the 
weight  of  the  stone  urges  it  incessantly  towards  the  surface  of  the 
earth.  Experience  proves  that  in  proportion  as  the  obstacles  to  a 
body's   motion  are  removed,  will  the  motion  itself  remain  unchanged. 

When  a  body  is  at  rest,  or  moving  with  uniform  motion,  its 
inertia   is   not   called  into   action. 

§29. — A  force  has  been  defined  to  be  that  which  changes  Oi^-teuds 
to__change  the  state  of  a  body  in  respect  to  rest  or  motion.  Weight 
and  Heat  are  examples.  A  body  laid  upon  a  table,  or  suspended 
from  a  fixed  point  by  means  of  a  thread,  would  move  under  the 
action  of  its  weight,  if  the  resistance  of  the  table,  or  that  of  the 
fixed  point,  did  not  continually  destroy  the  effort  of  the  weight.  A 
Dody  exposed  to  any  source  of  heat  expands,  its  particles  recede 
from  each  other,  and  thus  the  state  of  the  body  is  changed. 

3 


34      ELEMENTS  OF  ANALYTICAL  MECHANICS. 

When  we  push  or  pull  li  body,  be  it  free  or  fixed,  we  experience 
a  sensation  denominated  pressure,  traction,  or.  in  general,  ej^'ort.  This 
effort  is  analogous  to  that  which  we  exert  in  raising  a  weight.  Forces 
are  real  pressures.  Pressure  may  he  strong  or  feeble ;  it  therefore 
has  magnitude,  and  may  be  expressed  in  numbers  by  assuming  a 
certain  pressure  as  tniity.  The  unit  of  pressure  will  be  taken  to  be 
that  exerted  by  the  weight  of  -g^^g-  part  of  a  cubic  foot  of  distilled 
water,  at  38°,75,   and  is  called  a  2^otmd. 

1 30. — The  intoisity  of  a  force  is  its  greater  or  less  capacity  to 
produce  pressure.  This  intensity  may  be  expressed  in  pounds,  or  in 
quantity  of  motion.  Its  value  in  pounds  is  called  its  statical  mea- 
sure ;    in   quantity   of  motion,  its  dynamical   measure. 

I  31. — The  ^j>o<'?zi  of  a2iplicatioii  of  a  force,  is  the  material  point  to 
which  the  force  may  be  regarded  as  directly  applied. 

g32. — The  line  of  direction  of  a  force  is  the  right  line  which  the 
point  of  application  would  describe,  if  it  were  perfectly  free.  ■^  /W-i**^' 

g33. — The  effect  of  a  force  depends  upon  its  intensity,  point  of 
application,  and  line  of  direction,  and  when  these  are  given  the  force 
is  known. 

§34. — Two  forces  are  equal  when  substituted,  one  for  the  other, 
in  the  same  circumstances,  they  produce  the  same  effect,  or  when 
directly  opposed,  they  neutralize  each  other. 

§35. — There  can  be  no  action  of  a  force  without  an  equal  and 
contrary  reaction.  Ifhis  is  a  law  of  nature,  and  our  knowledge  of  it 
comes  from  experience.  If  a  force  act  upon  a  body  retained  by  a 
nxed  obstacle,  the  latter  will  oppose  an  eq\)al  and  contrary  resistance. 
If  it  act  upon  a  free  body,  the  latter  will  change  its  state,  and  in 
the  act  of  doing  so,  its  inertia  will  oppose  an  equal  and  contrary 
resistance.      Action  and   reaction   arc  ever   equal,  contrary  and  simidta- 

7ie07(S. 

§  30).-*— If  a  free  body  be  drawn  by  a  thread,  the  thread  m  ill  stretch 
and  even  break  if  the  action  be  too  violent,  and  this  will  the  more 
probably  happen    in    projiortion   as   the  body  is  more  massive.      If  a 


MECHANICS     OF     SOLIDS. 


bod}  be  suspended  by  means  of  a  vertical  chain,  and  a  wei^hin" 
spring  be  interposed  in  the  line  of  traction,  the  graduated  scale  of 
the  spring  will  indicate  the  weight  of  the 
body  when  the  latter  is  at  rest ;  but  if 
the  upper  end  of  the  chain  be  suddenly 
elevated,  the  spring  will  immediately  bend 
more  in  consequence  of  the  resistance 
opposed  by  the  inertia  of  the  body  while 
acquiring  motion.  When  the  motion  ac- 
quired becomes  uniform,  the  spring  will 
resume  and  preserve  the  degree  of  flexure 
which  it  had  at  rest.  If  now,  the  motion 
be  checked  by  relaxing  the  eflbrt  applied 
to  the  upper  end  of  the  chain,  the  spring  \ 

will   unbend    and    indicate  a  pressure   less  -^^^^ 

than  the  weight  of  the  body,  in  conse- 
quence of  the  inertia  acting  in  opposition  to  the  retardation.  The 
oscillati  nis  of  the  spring  may  therefore  serve  to  indicate  the  varia 
tions  in  the  motions  of  a  body,  and  the  energy  of  its  force  of 
inertia,  which  acts  against  or  with  a  force,  according  as  the  velo- 
city is  increased  or  diminished. 


§  37. — I'orces  produce  various  effects  according  to  circumstances. 
They  sometimes  leave  a  body  at  rest,  by  balancing  one  another, 
through  its  intervention ;  sometimes  they  change  its  form  or  break 
it ;  sometimes  they  impress  upon  it  motion,  they  accelerate  or  retard 
that  which  it  has,  or  change  its  direction  ;  sometimes  these  effects  are 
pr  iduced  gradually,  sometimes  abruptly,  but  however  produced,  they 
require  some  dejini/e  time,  and  are  etTected  by  covtinuoua  degrees.  If 
a  body  is  sometimes  seen  to  change  suddenly  its  state,  either  in 
respect  to  the  direction  or  the  rate  of  its  motion,  it  is  because  the 
force  is  so  great  as  to  produce  its  effect  in  a  time  so  short  as  to 
make  its  duration  imperceptible  to  our  senses,  yet  some  definite  jior- 
tion  of  time  is  necessary  for  the  change.  A  ball  fired  from  a  gun 
will  break  through  a  pane  of  glass,  a  piece  of  board,  or  a  sheet  of 
paper,  when  freely  suspended,  with  a  rapidity  so  great  as  to  call  int;C 


}6 


ELEMENTS  OF  ANALYTICAL  MECHANICS. 


action  a  force  of  inertia  in  the  parts  which  remain,  greater  than 
the  molecuUir  forces  which  connect  the  latter  with  those  torn  away, 
hi  such  cases  the  effects  are  obvious,  while  the  times  in  Avhich 
they  are  accomplished  are  so  short  as  to  elude  the  senses  :  and  yet 
these  times  have  had  some  definite  duration,  since  the  changes,  corres 
ponding  to  these  effects,  have  passed  in  succession  through  their  differ 
ent  degrees  from  the  beginning  to  the  ending. 

§  38. — Forces  which  give  or  tend  to  give  motion  to  bodies,  are 
called  motive  forces.  The  agent,  by  means  of  which  the  force  is 
exerted,   is  called  a  Motor. 

§  39. — The  statical  measure  of  forces 
may  be  obtained  by  an  instrument  called 
the  Dynamometer,  wdiich  in  principle  does 
not  differ  from  the  spring  balance.  The 
dynamical  measure  will  be  explained  fur- 
ther   on. 

§40. — When  a  force  acts  against  a  point 
in  the  surface  of  a  body,  it  exerts  a  pres- 
sure which  crowds  together  the  neighbDr- 
ing  particles  ;  the  body  yields,  is  compress- 
ed  and    its   surface  indented ;    the  crowded 

particles  make  an  effort,  by  their  molecular  forces,  to  regain  their 
primitive  places,  and  thus  transmit  this  crowding  action  even  to  the 
remotest  particles  of  the  body.  If  these  latter  particles  are  fixed  or 
prevented  by  obstacles  from  moving,  the  result  will  be  a  compression 
and  change  of  figure  throughout  the  body.  If,  on  the  contrary,  these 
extreme  particles  arc  free,  they  will  advance,  and  motion  will  be  com- 
municated by  degrees  to  all  the  parts  of  the  body.  This  internal  motion, 
the  result  of  a  series  of  compressions,  proves  that  a  certain  time  is 
necessary  for  a  force  to  produce  its  entire  effect,  and  the  error  of 
supposing  that  a  finite  velocity  may  be  generated  instantaneously. 
The  same  kind  of  action  will  take  place  when  the  force  is  employed 
to  destroy  the  motion  which  a  body  has  already  acquired ;  it  will 
first  destroy  the  motion  of  the  molecules  at  and  nearest  the  point  of 
action,  and  then,  by  degrees,  that  of  those  which  are  more  remote 
in  the  order  of  distance. 


MECHANICS     OF     SOLIDS.  37 

The  molecuhir  springs  cannot  be  compressed  without  reacting  in  a 
contrary  direction,  and  Avith  an  equal  eflbrt.  The  agent  which  presses 
a  body  will  experience  an  equal  pressure ;  reaction  is  equal  and  con- 
trary to  action.  In  pressing  the  linger  against  a  body,  in  pujling  it 
with  a  thread,  or  pushing  it  with  a  bar,  we  are  pressed,  drawn,  or 
pushed  hi  a  contrary  direction,  and  with  an   equal  effort.     Two  weigh- 


ing springs  attached  to  the  extremities  of  a  chain  or  bar,  will  indicate 
the  same  degree  of  tension  and  in  contrary  directions  when  made  to 
act  upon  each  other  through  its  intervention. 

In  every  case,  therefore,  the  action  of  a  force  is  transmitted  through 
a  body  to  the  ultimate  point  of  resistance,  by  a  series  of  ec[ual  and 
contrary  actions  and  reactions  which  balance  each  other,  and  which 
the  molecular  springs  of  all  bodies  exert  at  every  point  of  the  right 
line,  along  which  the  force  acts.  It  is  in  virtue  of  this  property  of 
bodies,  that  the  action  of  a  force  may  be  assumed  to  be  exerted  at 
any  'point  in   its  line  of  direction   ivithin   the  boundary  of  the  body. 

§41. — Bodies  being  more  or  less  extensible  and  compressible,  when 
interposed  between  the  motor  and  resistance,  will  be  stretched  or 
compressed  to  a  certain  degree,  depending  upon  the  energy  with  which 
these  forces  act;  but  as  long  as  the  force  and  resistance  remain  the 
same,  the  body  having  attained  its  new  dimensions,  will  cease  to 
change.  On  this  account,  we  may,  in  the  investigations  which  follow, 
assume  that  the  bodies  employed  to  transmit  the  action  of  forces  from 
one  point  to  another,  are  inextensiblo  and  rigid. 

WORK. 

§42.— To  tvork  is  to  overcome  a  resistance  continually  recurring 
along  some  path.  Tlius,  to  raise  a  body  through  a  vertical  height,  its 
weight  must   be   overcome    at    every  point  of  the  vertical  path.     If  a 


38  ELEMENTS     OF     AXALYTICAL     MECHANICS. 

body  fall  tln-ough  a  vertical  height,  its  -weight  develops  its  inertia  at 
every  point  of  the  descent.  To  take  a  shaving  from  a  board  with  a 
plr:ue,  the  cohesion  of  the  wood  must  be  overcome  at  every  point 
along  the  entire  length  of  the  path  described  by  the  edge  of  the  chisel. 

§43. — -The  resistance  may  be  constant,  or  it  may  be  variable.  In 
the  first  case,  the  quantity  of  work  performed  is  the  constant  resistance 
taken  as  many  times  as  there  are  points  at  which  it  has  acted,  and 
is  measured  by  the  product  of  the  resistance  into  the  path  described 
by  its  point  of  application,  estimated  in  the  direction  of  the  resistance. 
When  the  resistance  is  variable,  the  quantity  of  work  is  obtained  by 
estimatmg  the  elementary  quantities  of  work  and  taking  their  sum. 
By  the  elementary  quantity  of  work,  is  meant  the  intensity  of  the 
variable  resistance  taken  as  many  times  as  there  are  points  in  the 
indefinitely  small  path  over  which  the  resistance  may  be  regarded  as 
constant ;  and  is  measured  by  the  intensity  of  the  resistance  into  the 
difibrential  of  the  path,  estimated  in  the  direction  of  the   resistance. 

§  44. — In  general,  let  P  denote  any  variable  resistance,  and  «  the 
path  described  by  its  point  of  application,  estimated  in  the  direction 
of  the  resistance ;    then  will  the  quantity  of  work,  denoted  by   (2,  be 

given  by 

Q  =  /P.ds (T) 

which  integrated  between  certain  limits,  will  give  the  value  of  Q. 

§45. — The  simplest  kind  of  work  is  that  performed  in  raising  a 
weis;ht  through  a  vertical  height.  It  is  taken  as  a  standard  of  com- 
parison,  and  suggests  at  once  an  idea  of  the  quantity  of  work 
expended   in   any  particular   case. 

Let    the  weight   be   denoted   by    W,  and  the  vertical  height  by  11 ; 

till  11  v.- ill 

Q---W.H (8). 

If   W  become  one  pound,  and  //  one  foot,  then  will 

^=  1  ; 

and  the  unit  of  work  is,  therefore,  the  unit  of  force,  one  pound, 
exerted    over    the    unit  of  distance,  one  foot;    and    is   measured  by  a 


MECHANICS     OF    SOLIDS. 


39 


square  of   which  the  adjacent  sides  are  respectively  one  foot  and  one 
pound,  taken  from   the    same   scale  of  equal  parts. 

§  4G.— To  illustrate  the  use  of  Equation  (7),  let 
it  be  required  to  compute  the  quantity  of  work 
necessary  to  compress  the  spiral  spring  of  the 
common  spring  balance  to  any  given  degree,  say 
from  the  length  AB  to  DB.  Let  the  resistance 
vary  directly  as  the  degree  of  compression,  and 
denote  the  distance  AB'  by  x  ;    then  will 

F  =  C.x; 

in  which  C  denotes  the  resistance  of  the  spring 
when  the  balance  is  compressed  through  the  dis- 
tance unity. 

This  value  of  P  in    Equation  (7),  gives 

Q=  fP  .dx  =  j'C.  xdx  =  (7-  ^  +  C", 
which  integi'ated   betAveen  the  limits    x  =  0    and   x  =  AB  =  a.  gives 

Let   C=:10  pounds,  a  =  3  feet;  then  will 

Q  =  45     units  of  work, 

and  the  quantity  of  work  will  be  equal  to  that  required  to  raise 
45  pounds  through  a  vertical  height  of  one  foot,  or  on^:  pound, 
through  a  height  of  45  feet,  or  9  pounds  through  5  feet,  or  5 
pounds  through  9  feet,  &c.,  all  of  which  amounts  to  the  same  thing. 

§47. — A  mean  resistance  is  that  which,  multiplied  into  the  entire 
path  described  in  the  direction  of  the  resistance,  will  give  the  entire 
quantity  of  work.  Denote  this  by  i?,  and  the  entire  path  by  s, 
and  from    the   definition,  we  have 


■whence. 


E.s  =  fP.ds; 


(»)• 


40 


ELEMENTS     OF     ANALYTICAL     MECHANICS. 


That  is,  the  mean  resistance  is  equal  to  the  entire  work,  divided 
by  the    entire   path. 

In  the  above  example  the  path  being  3  feet,  the  mean  resistance 
would    be    15    pounds. 


j  §48. — Equation  (7)  shows  that  the  quantity  of  work  is  equal  to 
.the  area  included  between  the  path  s,  in  the  direction  of  the 
resistance,  the  curve  whose  ordinates  are  the  diiferent  values  of  P,  and 
the  ordinates  which  denote  the  extreme  resistances.  Whenever, 
therefore,  the  curve  which  connects  the  resistance  with  the  path  is 
known,  the  process  for  finding  the  quantity  of  work  is  one  of 
simple   integration. 

Sometimes   this    law   cannot    be   found,    and    the    intensity    of   the 
resistance   is   given   only  at  certain  points  of  the   path.     In  this  case 
we    proceed    as   follows,    viz.  :    At   the    several    points    of    the   path 
where    the   resistance   is    known,    erect    ordinates    equal    to    the    cor- 
responding   resistances,    and   connect    their     extremities    by    a    curved 
line  ;    then  divide   the  path  described  into   any  even  number  of  equal 
parts,  and  erect  the  ordinates 
at  the  points  of  division,  and 
at    the    extremities  ;    number 
the     ordinates     in    the    order 
of  the  natural  numbers;    add 
together    the  extreme   ordinates, 
increase   this  sum  by  four  times 
that  of  the  even  ordinates  and 
twice  that  of  the    uneven  ordi- 
nates, and  multiply  by  one-third 
of    Ihe    distance     between    any 
two   consecutive   ordinates. 

Demonstration :  To  compute  the  area  comprised  by  a  curve,  any 
two  of  its  ordinates  and  the  axis  of  abscisses,  by  plane  geometry, 
divide  it  into  elementary  areas,  by  drawing  ordinates,  as  in  the 
last  figure,  and  regard  each  of  the  elementary  figures,  e^  e.^  n  r^, 
e.i  e.,  r-i  r^,    &c.,    as    trapezoids  ;    it    is    ob\lous    that    the    error    of 


MECHANICS     OF    SOLIDS. 


41 


W2i  »i  +  e.^r 


this  supposition  will  be  less,  in  proportion 
as  the  number  of  trapezoids  between  given 
limits  is  greater.  Take  the  first  two  trape- 
zoids of  the  preceding  figure,  and  divide  the 
distance  e^  e^  into  three  equal  parts,  and  at 
the  points  of  division,  erect  the  ordinates 
7)1  n,  nil  ??!  ;  the  area  computed  from  the  three 
trapezoids  e^  m  n  7\,  m  mi /ij  n,  TOj  e^  7'3  ?ii,  will 
be  more  accurate  than  if  computed  from  the 
two  e,e.r.2r^,   e,e.^r.^r,. 

The   area  by  the    three    trapezoids  is 

e^  r^  +  m  n  m  n  +  ??(.,  n, 

^1  ^'^  ^  Ty ■  +  "^  "*i i^ +  "h  fa 

^  ^  2 

Bu'!   by   construction, 

6;  m  —  m  TO,  =  m,  c^  =  i  e,  e^  =  f  e,  e.^, 

and   the    above   may   be   written, 

1  fi  ^,2  (f,  r,  +  2  «i  ??.  +  2  ?/ii  ;(,  +  ^3  9-3), 

but    in    the   trapezoid  m  w,  ■»!  n, 

2  TO  n  +  2  ?»i  «i  =  4  Co  r,2,     very  nearly  ; 

whence    the   area   becomes 

i  c,  e.,  {e,  r,  +  4  c^  r,,  +  ^-g  j-g)  : 

the  area  of  the  next  two  trapezoi.ds  in  order,  of  the  preceding 
figure,  will   be 

i  e,  e,  [e^  7-3  +  4  64  ?-4  4-  e^  r,)  ; 

and  similar  expressions  for  each  succeeding  pair  of  trapezoids. 
Taking  the  sum  of  these,  and  we  have  the  whole  area  bounded  by 
the   curve,  its   extreme    ordinates,  and   the   axis    of  abscisses ;    or, 

Q  =  ^e,e,  [e,r,  +  4  e,r,  ■{-  2  e,  r,  f  4  r,r,  +  2e,r,  +  4  e,r,  +  e,r,']  .  (10) 
whence  the   rule. 


'r 


42      ELEMENTS  OF  ANALYTICAL  MECHANICS. 

§  49. — By  the  processes  now  explained,  it  is  easy  to  estimate  the 
quantity  of  work  of  the  weights  of  bodies,  of  the  resistances  due  to 
the  forces  of  affinity  which  hold  their  elements  together,  of  their 
elasticity,  &c.  It  remains  to  consider  the  rules  by  which  the  quantity 
of  woi'k  of  inertia  may  be  computed.  Inertia  is  exerted  only  during 
a  change  of  state  in  respect  to  motion  or  rest,  and  this  brings  us 
to    the  subject  of    varied    motion. 

VARIED    MOTION. 

,  / 

2  §  50. — Varied  motion  has  been  defined  to  be  that  in  which  unequal 
spaces  are  described  in  equal  successive  portions  of  time.  In  this 
kind  of  motion  the  velocity  is  ever  vaiying.  It  is  measured  at  any 
given  instant  by  the  length  of  path  it  would  enable  a  body  to 
describe  in  the  first  subsequent  unit  of  time,  were  it  to  remain 
unchanged.  Denote  the  space  described  by  s,  and  the  time  of  its 
description    by    t. 

However  variable  the  motion,  the  velocity  may  be  regarded  as 
constant  during  the  indefinitely  small  time,  dt.  In  this  time  the 
body  will  describe  the  small  space  ds  ;  and  as  this  space  is  des- 
cribed uniformly,  the  space  described  in  the  unit  of  time  would, 
were  the  velocity  constant,  be  ds  repeated  as  many  times  as  the 
unit  of  time  contains  dt.  Hence,  denoting  the  value  of  the  velo- 
city at   any   instant   by   v,    we    have 

v  =  dsx^^; 

or, 

ds 

-  =  * (") 

v  §51. — Continual  variation  in  a  body's  velocity  can  only  be  pro- 
duced by  the  incessant  action  of  some  force.  The  body's .  inertia 
opposes  an  equal  and  contrary  reaction.  This  reaction  is  directly 
i:)roportional  to  the  mass  of  the  body  and  to  the  amount  of  change 
in  its  velocity ;  it  is,  therefore,  directly  proportional  to  the  product 
of  the  mass  into  the  increment  or  dccrenu'nt  of  the  velocity.  The 
product  of  a   mass    into    a    velocity,    represents  a  quanlity  of  motion. 


^     ^,  MECHANICS     OF     SOLIDS.  43 

The  intensity  of  a  motive  force,  at  any  instant,  is  assumed  to  be 
measured  by  the  quantity  of  motion  which  this  intensity  can  generate 
in   a   unit  of  time. 

The  mass  remaining  the  same,  the  velocities  generated  in  equal 
successive  portions  of  time,  by  a  constant  force,  must  be  equal  to 
each  other.  However  a  force  may  vary,  it  may  be  regarded  as 
constant  during  the  indefinitely  short  interval  dt ;  in  this  time  it  will 
generate  a  velocity  dv^  and  were  it  to  remain  constant,  it  would 
generate  in  a  unit  of  time,  a  velocity  equal  to  dv  repeated  as  many 
times  as  dt  is  contained  in  this  unit;  that  is,  the  velocity  generated 
would   be   equal   to 


1        dv  ^/2J, 


dv .        

d(        dt 


and    denoting   the   intensity  of  the   force   by  P,  and  the  mass  by   M, 
we  shall  have 

P  =  3f/Il (12) 

dt 

Again,  differentiating  Equation  (11),  regarding  t  as  the  indej^endent 
variable,  we  get, 

dv    —    —r--. 

dt  ' 

and    this,    in    Equation    (12),    gives 

P  =  J/.ff.     . (13) 

From  Equation  (11),  we  conclude  that  in  varied  motion,  the  velocity 
at  any  instant  is  equcd  to  the  Jirst  differential  co-efficient  of  the  space 
rcyarded  as  a  function  of  the  time. 

From  Equation  (12),  that  the  intensity  of  any  motive  force,  or  of 
the  inertia  it  develops,  at  any  instant,  is  measured  by  the  product  of 
tlie  mass  into  the  first  differential  co-cffcient  of  the  velocity  regarded  as 
a  function  of  the  time. 

And  from  Equation  (13),  that  the  intensity  of  the  motive  force  or 
of  inertia,  is  measured  by  the  product  of  the  mass  into  the  s-cond 
diffei'cnfial  co-ffficient  of  the  space  regarded  as  a  function  of  the  time. 


44  ELEMENTS     OF    ANALYTICAL    MECHANICS. 

§  52. — To  illustrate.     Let  there  be  the  relation 

s  =  aP  +  bfi (14) 

required  the  space  described  in  three  seconds,  the  velocity  at  the  end 
of  the  third  second,  and  the  intensity  of  the  motive  force  at  the  s;inie 
instant. 

Differentiating  Equation  (14)  twice,  dividing  each  result  by  dt.  and 
multiplying  the  last  by  M.  we  fuid 

^  =  v  =  ?Mtr-  +  2ht   ■    ■   ■    '    (15) 

M—^P^MlCyat  +  ^h]'     .     •     (16) 

]\fake  a  =  20  feet,  6  =  10  fcL't,  and  t  =  3  seconds,  we  have, 
from  Equations  (14),  (15),  and    (IG), 

s  r=  -20  .  33  +  10  .  32  =  030   feet ; 

V  =  3  .  20  .  3-  +  2  .  10  .  3  ==  COO    feet ; 

F  =  J/(G  .  20  .  3  +  2  .  10)  =  380  .  J/. 

That  is  to  say,  the  body  will  move  over  the  distance  630  feet  in 
three  seconds,  will  have  a  velocity  of  600  feet  at  the  end  of  the 
third  second,  and  the  force  will  have  at  that  instant  an  intensity 
capable  of  generating  in  the  mass  J/,  a  velocity  of  380  feet  in  one 
second,  were  it  to  retain  that  intensity  unchanged. 

§53. — Dividing    Equations    (12)    and    (13)    by    J/,  they   give 

(1^) 


F 
M~ 

dv 
dt 

F 

J2.9 

M~ 

dfi 

(18) 


The  first  member  is  the  same  in  both,  and  it  is  obviously  that 
jjortion  of  the  force's  intensity  which  is  impressed  upon  the  unit  of 
mass.  The  second  member  in  each  is  the  velocity  impressed  in  the 
iniit  of  time,  and  is  called  the  acceleration  due  to   the  motive  fo''ce. 


MECIIANICSOFSOLIDS.  45 

§54. — From    Equation    (11)   we   have, 

ds  =  V  .dt (19) 

multiplying  this  and  Equation   (12)  together,  there  will  result, 
P  .ds  —  M.v  .du       .     .     .     .     (20) 


and   integrating, 


fP.ds=  — ;- (21) 


The  first  member  is  the  quantity  of  work  of  the  motive  force, 
which  is  equal  to  that  of  inertia ;  the  product  M.  v^,  is  called  the 
living  force  of  the  body  whose  mass  is  M.  Whence,  we  see  that 
ihe  u'ork  of  inertia  is  equal  to  half  the  living  force  ;  and  the  living 
force  of  a  body  is  double  ihe  quantity  of  tvork  expended  hy  its  inertia 
u'hile  it  is  acquiring  its  velocity. 

§  55. — If  the  force   become   constant  and  equal  to  i^,   the  motion 
will    be    uniformly    varied,    and    we    have,    from    Equation    (18), 

F  _  d^s 
M  ~  Ifi 

Multiplying  by  dt  and  integrating,  we  get 


^ 


|-'  =  ,t+^'  =  "  +  ^  •  •  (-) 


and  if  the   body  be  moved  from   rest,  the  velocity    will    be  equal  to 
zero  when  t  is  zero ;    whence   C  =  0,  and 

|-^  =  - (23) 

^Multiplying  Equation    (22)    by    dt,    after  omitting   G   from  it,  and 
integrating  again,  we  find 

and  if  the  body  start  from  the  origin  of  spaces,  C  will  be  zero,  and 
F    t"^ 


46  ELEMENTS     OF    ANALYTICAL    MEGHAN.  :S. 

ISIaking    t  equal    to    one    second,  in   Equations    (24)    and  (23),  and 
dividing  the  last  by   the  first,  we  have 

2.  _  i. 

2"  ~  7' 

or,  V  =  2s (25) 

That  is  to  say,  the  velocitij  generated  in  the  first  unit  of  time  is 
measured  by  double  the  space  described  in  acquiring  this  velocity. 
Equations   (23),    (24),  and   (25)    express   the  laws   of  constant  forces. 

g  5(5. — The  dynamical  measure  for  the  intensity  of  a  force,  or  the 
pressure  it  is  capable  of  producing,  is  assumed  to  be  the  effect  this 
pressure  can  produce  in  a  unit  of  time,  this  effect  being  a  quantity 
of  motion,  measured  by  the  product  of  the  mass  into  the  velocity 
generated.  This  assumed  measure  must  not  be  confounded  with  the 
quantity  of  work  of  the  force  while  producing  this  effect.  The 
former  is  the  measure  of  a  single  pressure;  the  latter,  this  pressure 
repeated  as  many  times  as  there  are  points  in  the  path  over  which 
this   pressure   is    exerted. 

Thus,  let  the  body  be  moved  from  A  to 
B,  under  the  action  of  a  constant  force,  in 
one  second ;  the  velocity  generated  will. 
Equation  (25),  be  2AB.  Make  I]C=2AB, 
and  complete  the  square  BCFE.  BE  will 
be  equal  to  v ;  the  intensity  of  the  force 
will  be  M.v;  and  the  quantity  of  work, 
the  product  of  M.v  by  AB,  or  by  its 
equal  \  v ;  thus  making  the  Cjuantity  of 
work  -I  M  v^^  or  the  mass  into  one  half  the 
square  BF\  which  agrees  with  the  result  obtained  from  Equation  (2i). 


r 


EQUILIBRroM. 


1 57, — Equilibrium  is  a  term  employed  to  express  the  state  of 
two  or  more  forces  which  balance  one  another  through  the  interven- 
tion of  some  body  subjected  to  their  simultaneous  action.  "When 
applied   to  a  body,  it  means  that  the  body  is  at  rest. 


MECHANICS     OF     SOLIDS.  47 

Wc  must  be  careful  to  distinguish  between  the  extraneous  forces 
which  act  upon  a  body,  and  the  forces  of  inertia  which  they  niay,  or 
may   not,  develop. 

If  a  body  subjected  to  the  simultaneous  action  of  several  extraneous 
forces,  be  at  rest,  or  have  uniform  motion,  the  extraneous  forces  are 
in  equilibrio,  and  the  force  of  inertia  is  not  developed.  If  the  body 
have  varied  motion,  the  extraneous  forces  are  not  in  equilibrio,  but 
develop  forces  of  inertia  which,  with  the  extraneous  forces,  are  in 
equilibrio.  Forces,  therefore,  including  the  force  of  inertia,  are  ever 
in  equilibrio  ;  and  the  indication  of  the  presence  or  absence  of  the 
force  of  inertia,  in  any  case,  shows  that  the  body  is  or  is  not  chang 
ing  its  condition  in  respect  to  rest  or  motion.  This  is  but  a  conse- 
quence of  the  universal  law  that  every  action  is  accompanied  by  an 
equal  and  contrary  reaction. 

THE    COED. 

I  58. — A  cord  is  a  collection  of  material  points,  so  united  as 
to  form  one  continuous  and  flexible  line.  It  will  be  considered, 
in  what  immediately  follows,  as  perfectly  flexible,  inextensible,  and 
without    thickness    or   icei(jht. 

§  ,59. — By  the  tension  of  a  cord  is  meant,  the  effort  by  which  any 
two  of  its    adjacent  particles  are  urged  to  separate  from  each  other. 

S  60. — Two    equal    forces,  F  and    P^,  applied    at    the    extremities 
A,  A'  of  a  straight  cord,  and 
acting  in   opposite  directions 

from    its    middle    point,  will  S^ -^ ■,„r,7rrr'r^,,.7,7Tn^. .-T 

0 

maintain  each  other  in  equi- 
librio.     For,    all    the    points 

of  the  cord  being  situated  on  the  line  of  direction  of  the  forces,  any 
one  of  .them,  as  0,  may  be  taken  as  the  common  point  of  applica- 
tion without  altering  their  elfects ;  but  in  this  case,  the  forces  being 
equal  will,    §34^    neutralize    each   other. 


4:8 


ELEMENTS     OF     ANALYTICAL    MECHANICS. 


§61. — If  two    equal  forces,  P  unci  P',  solicit  in  opposite  directions 

the    extremities   of  the   cord 

A  A^,  the  tesision  of  the  cord 

Avill  Ijg  measured  by  the  in-  jp'  ^ A  p 

J  . ^ — ^  > 

tensity  of  one  of    the  forces. 
For,  the    cord    being  in  this 

case  in  equilibrio,  if  we  suppose  any  one  of  its  points  as  0,  to  become 
fixed,  the  equilibrium  will  not  be  disturbed,  while  all  communica- 
tion between  the  forces  will  be  intercepted,  and  either  force  may 
be  destroyed  without  aftecting  the  other,  or  the  part  of  the  cord  on 
which  it  acts.  But  if  the  part  A  0  of  the  cord  be  attached  to  a 
fixed  point  at  0,  and  draAvn  by  the  force  P  alone,  this  force  must 
measure   the    tension. 


TIEE    MUFFLE. 
-3  -4 

§  62. — Suppose  A,  A%  B,  B\  &c.,  to  be  several  small  wheels  or 
pulleys  perfectly  free 
to  move  about  their 
centres,  which,  con- 
ceive for  the  present 
to  be  fixed  points. 
Let  one  end  of  a  cord 
be  fastened  to  a  fixed 
point  C,  and  be 
Avound     around     the  CE% 

pulleys  as  represent- 
ed in  the  figure ;  to  the  other  extremity,  attach  a  weight  w.  The 
weight  ru  will  be  maintained  in  equilibrio  by  the  resistance  of  the 
fixed  point  C,  through  the  medium  of  the  cord.  The  tension  of  the 
cord  will  be  the  same  throughout  its  entire  length,  and  equal  to 
the  v.-eight  w  ;  for,  the  cord  being  perfectly  flexible,  and  the  wheels 
perfectly  free  to  move  about  their  centres,  there  is  nothjng  to 
intercept  the  free  transmission  of  tension  from  one  cud  to  the  other. 

Let   the   points  s   and   r  of  the   cord   be  supposed  for    a   moment, 
fixed  ;  the   intermediate  portion  s  r  may  be  removed  without  aflfecting 


MECHANICS     OF     SOLIDS.  49 

the  tension  of  the  cord,  or  the  equilibrium  of  the  weight  w.  At 
the  i^oint  r,  apply  in  the  direction  from  r  to  a,  a  force  whose  inten- 
sity is  equal  to  the  tension  of  the  cord,  and  at  s  an  equal  force 
acting  in  the  direction  from  s  lo  h\  the  points  r  and  s  may  now  be 
regarded  as  free.  Do  the  same  at  the  points  s',  r',  s'\  r'\  s'"  and 
r'"^  and  the  action  of  the  weight  w,  upon  the  pulleys  A  and  A'  will 
be  replaced  by  the  four  forces  at  s,  s^,  a"  and  s"\  all  of  equal  in- 
tensity and  acting   in    the   same   direction. 

Now,  let  the  centres  of  the  pulleys  A  and  ^'  be  firmly  con- 
nected with  each  other,  and  with  some  other  fixed  point  as  W2,  in 
the  direction  of  BA  produced,  and  suppose  the  pulleys  diminished 
indefinitely,  or  reduced  to  their  centres.  Each  of  the  points  A  and 
A'  will  be  solicited  in  the  same  direction,  and  along  the  same  line, 
by  a  force  equal  to  ^2w^  and  therefore  the  point  ??i,  by  a  force 
equal    to    4r<;. 

Had  there  been  six  pulleys  instead  of  four,  the  point  m  would 
have  been  solicited  by  a  force  equal  to  0^y,  and  so  of  a  greater 
number.  That  is  to  say,  the  point  m  would  have  been  solicited  bv 
a  force  equal  to  w^  repeated  as    many  times  as  there  are  pulleys. 

If  the  extremity  C  of  the  cord  had  been  connected  with  the  point 
?K,  after  passing  round  a  fifth  pulley  at  (7,  the  point  m  would 
have  been  subjected  to  the  action  of  a  force  equal  to  ^lo  \  if 
seven  pulleys  had  been  employed,  it  v/ould  have  been  urged  by  a 
force  Iw ;  and  it  is  therefore  apparent,  that  the  intensity  of  the 
force  which  solicits  the  point  v/i,  is  found  by  multiplying  the  tension 
of  the  cord,  or  weight  iv,  hy  the  number  of  -pulleys. 

This  combination  of  the  cord  with  a  number  of  wheels  or  pulleys, 
is  called  a  muffle. 

§  Go. — Conceive  the  point  m  to  be  transferred  to  the  position 
in'  or  ni",  on  the  line  AB.  The  centres  of  the  pulleys  A,  A',  &c., 
being  invariably  connected  with  the  point  m,  will  describe  equal 
paths,  and  each  equal  to  m  m',  or  m  m",  so  that  each  of  the  parallel 
portions  of  the  cord  will  be  shortened  in  the  first  case,  or  length- 
ened in  the  second,  by  equal  quantities;  and  if  e  denote  the  length 
of  the   path    described   by  to,    n  the   number  of  parallel  portions  of 

3 


50 


ELEMENTS     OF    ANALYTICAL    MECHANICS. 


the  cord,  which  is  equal  to  the  number  of  jDulleys,  and  |,  the  change 
in  length  of  the  portion  ^iiu  in  consequence  of  the  motion  of  m, 
we  shall  have,  because  the  entire  length  of  the  cord  remains  the  same, 

n.e  =  1 (26) 

The  first  member  of  this  equation  we  shall   refer  to  as  the  change 
in  length  of  cord  on  the  2^ulleys. 

3  §^- — The  action  of  any  force  P,  upon  a  material  point,  may  be 

replaced  by  that  of  a  niuftle,  by  making  the  tension  of  its  cord  equal 
to  the  intensity  of  the  given  force^  divided  by  the  number  of  parallel 
portions  of  the  cord,  or  number  of  pullies. 

EQUILIBRIUM    OF    A    RIGID    SYSTi;^ 

^  §  G5. — Let  M  represent  a  collection  of  material  points,  united  in 
any  manner  whatever,  forming  a  solid  body,  and  subjected  to  the 
action  of  several  forces,  P,  P\  P",  P'",  &c.  ;  and  suppose  these 
forces  in  equilibrio. 

Find    the    greatest   force  w,  which   will   divide   each  of  the   given 
forces  without  a  remainder ;   replace  the  force  P  by  a  miiffl,e,   having 


a  number  of  pulleys   denoted   by  —  ;    the   tension  of  the  cord   will 


MECHANICS     OF    SOLIDS.  51 

be  denoted  hy  w.  Do  the  same  for  each  of  the  forces,  and  we 
shall  have  as  many  muffles  as  there  are  forces,  and  all  the  cords 
will   have   the  same    tension. 

Lot  the  several  cords  be  united  into  one,  as  represented  in  the 
figure,  one  end  being  attached  at  C,  the  other  acted  upon  by  a  weight 
equal  to  the  force  iv.  The  action  upon  the  body  will  remain  un- 
changed; that  is,  the  substituted  forces,  including  w,  will  be  in  equi- 
librio. 

In  this  state  of  the  system,   let  a  force   Q  be   applied  to  put   the 
body*in    motion,    and    at    the    instant   motion    begins,    withdraw    this 
force  and  stop  the  motion  before  the  equilibrium  of  the  forces  is  des 
troyed.     The  points  of  application  of 
the    original    foi'ces    will    each  have    .t^jz^,^' 
described  an  indefinitely  small  path,  't  ^  /-'' 
as  71111.      Let  mrhe  the  projection 
of  this  path  upon  the  original  direc- 
tion of  the    force,  and    denote   the 
length  of  this  projection  by  e.     Join 
the  point  ?i  with   any    point    o,   on 
the    direction    of   the    force   and    at 
some  definite  distance  from  7n.     From  the  triangle  onr,  we  have 

— 2         — 2         — 2 
on    ^  or    -\-  nr    ; 

^he   displacement   being    indefinitely    small,   nr     may    be   neglected   in 

2 

comparison  with  or  ,  being  an  indefinitely  small  quantity  of  the  second 
order ;    hence, 

on  =  0?*, 
and, 

om  —  on  =:  om  —  or  =  e. 

But  the  number  of  pulleys  in  the  muffle  which  acts  along  the 
direction  of  the   force  P   is, 

P  . 

5 

hence,  the   change  in   the  length  of  the   cord   on  the  pulleys  of  this 


52  ELEMENTS     OF     ANALYTICAL    MECHANICS. 

muffle,  caused  by  the  sliglit  motion  of  the  point  of  application  of  the 
force   P,  will,   since  the  centre  of  the  pulley  B   is  fixed,  be 

and  denoting  by  e',  e",  e'",  &c.,  the  projections  of  the  paths  described 
by  the  points  to  which  the  forces  P\  P" ^  P"\  &c.,  are  respectively 
applied,  on  the  original  directions  of  these  forces,  we  shall  have 

P'.e'      P".e"      P'^'.e'"    ,  • 

■ 5     ;      •  1    &C., 

IV  IV  W 

for  the  corresponding  changes  in  the  length  of  the  cord  on  the  other 
muffles. 

Jn  all  these  changes,  the  cord  being  inextensible,  its  entire  length 
remains  the  same,  and  if  the  change  in  length  which  the  portion  uiv 
undergoes    be  denoted  by  f,  we  shall  have 

-^  (P.c  +  P'.e'  +P".e"  +  i"".e"'  +  &c.)  +  g  =  0     .     .(27) 

This  equation  expresses  the  algebraic  sum  of  all  the  changes  m 
the  lengths  of  the  several  parts  of  the  cord,  between  the  points  of 
application,  and  the  fixed  points  towards  which  the  points  of  applica- 
tion are  solicited  ;  the  effect  of  these  changes  being  to  shorten  some 
and  lengthen  others,  some  of  the  terms  of  Equation  (27)  must 
be  negative. 

,  Now  it  is  one  of  the  essential  properties  of  a  system  of  forces 
in  eqnilibrio,  to  leave  a  body  subjected  to  their  action  as  free  to 
move  as  though  these  forces  did  not  exist.  The  additional  force  Q, 
therefore,  was  wholly  employed  in  developing  the  inertia  of  the 
body  M\  it  was  neither  assisted  nor  opposed  by  the  forces  repre- 
fKjntcd  by  the  action  of  the  nnifflcs,  because  these  forces  balanced 
each  other,  and  the  motion  was  arrested  before  the  points  of  appli- 
cation were  sufficiently  disturbed  to  break  up  the  equilibrium  ;  nor, 
reciprocally,  §  35,  was  the  action  of  the  muffles,  nor  the  tension  of 
the    cord    which    produced    this    action,    affected    by    Q.      Hence    the 


MECHANICS    OF    SOLIDS.  53 

tension  of  the  coid  was  invariable  dui-ing  the  disturbance.  But  an 
invariable  tension  must  have  kept  the  weight  w  at  rest  durinij  the 
displacement,  and  we  have 

?  —  0 

and  Equation  (27)  will  reduce  to, 

Pe  +  P'e'  +  P"e'    +  P"'e"'  +  &c.  ==  v-)  ;     ,     .     .     .    (2S) 

§  QQ). — It  may  be  objected,  that  the  given  forces  are  incommcnsn- 
rable,  and  that  therefore,  a  force  cannot  be  found  which  will  divide 
each  without  a  remainder;  to  Avhich  it  is  answered,  that  Equation 
(28),  being  perfectly  independent  of  the  value  of  the  weight  «•,  or 
tension  of  the  cord,  this  weight  may  be  taken  so  small  as  to  rendei 
the  remainder  after  division  in  any  particular  ease,  perfectly  inappre- 
ciable. 

§  67. — The   indefinitely    small    paths    m  n,  m'u',    described    by    the 
points  of  application  of  the  forces,  P  and  P',  during  the  slight  motion 
we  have  supposed,  are  called  virtual  veloci- 
ties ;    and  they  are  so   called,  because,  being 
the    actual    distances    passed    over    by    the  /\ 

points  to    which   the   forces    are   applied,  in        ^'L'^ 

the   same    time,   they    measure   the   relative  •       r'^f^ **-?' 

rates  of  motion   of  these  points.      The"  dis- 
tances r  m  and   r'ln',   represented    by  e  and 

e',  are  therefore,  the  projections  of  the  virtual  velocities  upon  the 
directions  of  the  forces.  These  projections  may  fall  on  the  side 
towards  which  the  forces  tend  to  urge  those  points,  or  the  reverse, 
depending  upon  the  direction  of  the  motion  imparted  to  the  system. 
In  the  first  case,  the  projections  are  recfardcd  as  2^osiiive,  and  in  the 
second,  as  negative.  Thus,  in  the  case  taken  for  illustration,  »i  r  is 
positive,  and  m'r'  negative.  The  products  Pe  and  P'e',  are  called 
virtual  moments.  They  are  the  elementary  quantities  of  work  of  the 
forces  P  and  P\  The  forces  are  always  regarded  as  positive  ;  the 
sign  of  a  virtual  moment  will,  therefore,  depend  upon  that  of  the 
projection   of  the  virtual   velocity. 

§08. — Referring  to  Equation  (28),  we  conclude,  therefore,  that  tvhen- 
ever  several  forces    are    in   eguilibrio,   the  atgelnaic  sum  of  their  virtual 


5i         jclements   of   analytical   mechanics. 

moments  is  equal  to  zero ;   and  in  this  consists  Avhat  is  called  the  prin 
ciple  of  virtual  velocities. 

§69. — Conversely,  if  in  any  system  of  forces,  the  algebraic  sum 
of  the  virtual  moments  be  equal  to  zero,  the  forces  avIU  be  in  cqui- 
librio.  For,  if  they  be  not  in  equilibrio,  some,  if  not  all  the  points 
of  application  will  have  a  motion.  Let  q^  q',  q'\  &;c.,  be  the  pro- 
jections of  the  paths  which  these  points  describe  in  the  first  instant 
of  time,  and  §,  Q\  Q'\  &c.,  the  intensities  of  such  forces  as  will, 
when  applied  to  these  points  in  a  direction  opposite  to  the  actual 
motions,  produce  an  equilibrium.  Then,  by  the  principle  of  virtual 
velocities,  we  shall  have 

Pe.  +  P'e'  4-  P"e"  +  &c.  +   Qq  -\-   Q'q'  +   Q"q"  +  &c.  =  0 

But  by  hypothesis, 

Pe  +  P'e'  +  P"e"  +  &c.  =  0, 
and  hence, 

Qq  +  Q'q'  +  Q"q"  +  &c.  =  0  .     .     .     (28)' 

Now,  the  forces  Q,  Q\  §'''',  &c.,  have  each  been  applied  in  a  direc- 
tion contrary  to  the  actual  motion  ;  hence,  all  the  virtual  moments  in 
Equation  (28)'  will  have  the  negative  sign  ;  each  term  must,  therefore, 
be  equal  to  zero,  which  can  0)ily  be  the  case  by  making  Q,  Q\  Q"^ 
&c.,  separately  equal  to  zero,  since  by  supposition  the  quantities 
denoted  by  q^  q',  q".  are  not  so.  We  therefore  conclude,  that  when 
the  algebraic  sum  of  the  virtual  moments  of  a  system  of  forces  is 
equal  to  zero,  the  forces  will  be  in  equilibrio. 

"Whatever  be  its  nature,  the  effect  of  a  force  will  be  the  same  if 
we  attribute  its  effort  to  attraction-  between  its  point  of  application 
and  some  remote  point  assumed  arbitrarily  and  as  fixed  upon  its  line 
of  direction,  the  intensity  of  the  attraction  being  equal  to  ihat  of  the 
for.v..  Denote  the  distance  from  the  point  of  application  of  P.  to 
that  towards  which  it  is  attracted,  by  p^  and  the  corresponding  dis- 
tances in  the  case  of  the  forces  /",  P",  &c.,  by  ;/,  jo",  &c.,  respect- 
ively ;  also,  let  o/>,  Sp\  Sp",  &zc.,  represent  the  augmentation  or  dimi- 
nution of  these  distances  caused  by  the  displacement,  supposed  indefj- 
nitely  small,  then  Fj  0."),  will 

€  =  op,  e'  —  Sp\  e"  =  oj/^,  <fcc., 


MECHANICS     OF    SOLIDS.  55 

aiid   Equation  (28)   may  be  written 

PSjy  +  P'Siy'  +  P"op"  +  &c.  =  0    .     .     .  (29)        j^^nry^ 

in  which  the  Greek  letter  6  simply  denotes  change  in  the  value  of 
the  letter  written  immediately  after  it,  this  change  arising  from  the 
small   displacement. 

§  TO. — If  the  extraneous  ftn'ccs  applied  to  a  body  be  not  in  eqni- 
librio,  they  will  communicate  motion  to  it,  and  will  develop  forces  of 
inertia  in  its  various  elementary  masses  with  which  they  will  be  in 
equilibrio  ;  and  if  extraneous  forces  equal  in  all  respects  to  these  forces 
of  inertia  were  introduced  into  the  system,  the  algebraic  sum  of  the 
virtual  moments  would  be   equal   to  zero. 

But  if  m  denote  the  mass  of  any  element  of  the  body,  s  the 
path  it  describes,  its  force  of  inertia  will,  Eq.  (13),  be 

m : 

and  denoting  the  projection  of  its  virtual  velocity  on  s  by  ^s,  its  vir- 
tual moment  will  be 

and  because  the  forces  of  inertia  act  in  opposition  to  the  extraneous 
forces,  their  virtual  moments  must  have  signs  contrary  to  those  of 
the  latter,  and  Equation  (29)  may  be  written 

2P.  fo  -  2wi .  '!!f  .  (Js  =  0 :  .     .     .     .  (30).       , 

in  which  2  denotes  the  algebraic  sum  of  the  terms  similar  to  that 
written  immediately  after  it. 

PRINCIPLE    OF    D'ALEMBERT. 

§71. — This  simple  equation  involves  the  whole  doctrine  of  Mechanics. 
The  extraneous  forces  P,  P',  P",  &c.,  are  called  impressed  forces. 
The  forces  of  inertia  M'hich  they  develop  may  or  may  not  be  equal  to 
them,  depending  upon  the  manner  of  their  application.  If  the  impressed 
forces  be  in  equilibrio,  for  instance,  they  will  develop  no  force  of  inertia; 


56  ELEMENTS     OF    ANALYTICAL    MECHANICS. 

but  in  all  cases,  the  forces  of  inertia  actually  developed  M-ill  be  equal 
and  contrary  to  so  much  of  the  impressed  forces  as  determines  the 
change  of  motion.  The  portions  of  the  impressed  forces  which  detei'- 
mine  a  cltmge  of  motion  are  called  effective  forces  ;  and  from  Ecpiatif>n 
(30),  we  infer  that  the  impressed  and  effective  forces  are  always  in  equi- 
librio  when  the  directions  of  the  latter  are  reversed,  uliirl  iiill  jnni 
-vTTir  Ull  ■■ehongo' r(;>£_B»Qjaa«..  This  is  usually  known  as  D'Alemhert's 
Fnnci;ple,  and  is  nothing  more  than  a  plain  consequence  of  the  law 
that  action  and  reaction   are   ever   equal  and  contrary. 

This  same  principle  is  also  enunciated  in  another  way.  Since  the 
effective  forces  reversed  would  maintain  the  impressed  forces  in  equi- 
librio,  and  prevent  them  from  producing  a  change  of  motion,  it 
follows  that  whatever  forces  may  he  lost  and  gained  must  be  in  equili- 
hrio ;  else  a  motion  different  from  that  which  actually  takes  place 
must  occur,   a  supposition  which  it  were  absurd  to  make. 

S  §  'i'2. — First  Transformation.     Equation  (30)  is  of  a  form  too  general 

for  easy  discussion.  To  transform  it,  refer  the  directions  of  the  forces 
and  their  points  of  a^^plication  to  three  rectangular  axes. 

Denote  by  «,  /3,  y,  the  angles  which  the  direction  of  the  force 
P  makes  with  the  axes  .r,  y,  £•,  respectively  ;  by  a,  6,  c,  the  angles 
which  its  virtual  velocity  makes  with  the  same  axes  ;  and  by  (p,  .the 
angle  which  the  virtual  velocity  and  direction  of  the  force  make  with 
each  other,   then  will 

SS' .  ^l^^.  ^^)  cos  9  z=z  cos  a  .  cos  a  +  cos  h  .  cos  /3  +  cos  c  .  cos  y. 

Denote  by  ^,  the  virtual  velocity,  and  multiply  the  above  equation 
by  Pk\  and  Ave  have 

Pk  cos  <p  —  Pk  cos  a  .  cos  a  +  Pk  cos  b  .  cos  (3  +  Pk  cos  c .  cos  y ; 

But  denoting  the  co-ordinates  of  tlic  point  of  application  of  P  by 
X,  y,  2,  we  have 

'    '.  k  cos  cp  =1  Sp  ;    k  cos  a  =  Sx  ;    k  cos  b  ^=i  6y  \    k  cos  c  =r  ^2  ; 

and  these  values  substituted  above,  give 

P  .Sp  =  P  cos  a  .o^x  +  P  cos  ft  .  oy  +  P  cos  y  .Sz.    .    .  (31 ). 

Similar  values  may  be  found  for  the  virtual  moments  of  other  forces. 


MECHANICS     OF    SOLIDS.  57 

§  T3. — If  P  bo  replaced  by  the  force  of  inertia,  tlien  will  a,  /3,  and 
y  denote  the  inclinations  of  the  direction  of  this  force  to  the  axes  xtjz\ 
k  its  virtual  velocijy ;  S  ^' -^\^f^  thy  inclinations  of  the  latter  to  the 
axes,  and  0  its  inclination  to  the  direction  of  the  force  of  inertia,  aiid 
we  may,  Eq.  (13),  write  ^  ^      '    j 

m'^—-kcoS(f,  =  m  — -  •  cos  «,A:  cos  «  +  )/(—-  cos  /3 .  /j  cos  6  +  t.i  -—  cos  y . k  cos  c. 
at-  ^  dt-  dt-  dt- 

-r.    .  *^    <^*^'  ^.' 

h  cos  0  =:  ds ;      A-  cos  «  =  6x, ;      A;  cos  b  =  dy]       k  cos  c  =  6:: ; 
(Z-5 .  cos  a.  =  d}x\     (Ps  cos  /3  =  d-y ;     cZ'^s  cos  y  =  d'^z  ; 
whence, 


7 "'  ='r  •  ^/  ^-^  +  "^  •  ^  ■  '^  +  "^  •  rf^  •  ^-^  ■'  •  (•'■^)' 


and  similai"  expressions  may  be  found  for  the  virtual  moments  of  ihe 
forces  of  inertia  of  the  other  elementarj^  masses. 

§  74. — If  the  intensity  of  the  force  P,  be  represented  by  a  pr,rtion 
of  its  line  of  direction,  which  is  the  practice  in  all  geometrical 
illustrations  of  Mechanios,  the  factors  P  cos  a,  P  cos  /3,  and  /■'  co:>  7, 
in  Equation  (31),  would  represent  the  intensities  of  forces  equal  to 
the  projections  of  the  intensity  P,  on  the  axes ;  and  regarding  these 
as  acting  in  the  directions  of  the  axes,  the  factors  ^.r,  Sy,  and  5z,  will 
represent  the  projections  of  their  virtual  velocities,  which  virtual  veloci- 
ties will  coincide  with  that  of  the  force  P. 
Again,  Equation  (32), 

d'^x  dh/  d^z 

dt^  dfi  dt^ 

are  forces  of  inertia  in  the  directions  of  the  axes,  and  Sx,  Sy,  Sz,  are 
the  projections  of  their  virtual  velocities ;  these  virtual  velocities  coincide 
•with  that  of  the  inertia  of  w. 

The    values  of   these  virtual   velocities   depend   upon   the  nature  of 
the  displacement. 


68 


ELEMENTS    OF    ANALYTICAL    MECHANICS, 


FREE    MOTION    OF    A    RIGID    SYSTEM. 

"7         §  15. — Second  Transformation.     By  the  substitution,  in  Equation  (30), 

d'^s 
for  P  dj)  and  m  .  y-^  .  6  s,  tlieir  values  in  Equations  (31)  and  (32),  there 

■would  result  an  equation  containing,  in  general,  three  times  as  many  vari- 
ations of  re  y  2  as  there  are  extraneous  forces  and  elementary  masses,  m. 
Where  the  forces  are  applied  to  a  body  whose  elementary  masses  are  in- 
variably connected — that  is,  to  a  rigid  solid — the  number  of  these  varia- 
tions is  greatly  reduced,  in  consequence  of  the  relations  determined  by 
this  connection. 

The     most    general   motion    we    can    attribute    to   a    body    is   one 
of   translation     and    of    rotation     combined.       A    motion    of    transla-- 
tion    carries    a    body    from    place    to    place    through    space,    and    its 
position,    at   any   instant,    is   determined   by   that   of  some  one    of  its 
elements.       A    motion    of    rotation   carries    the   elements    of    a   body 
around    some    assumed 
point.      In  this  investi- 
gation,   let    this    point 
be     that    which     deter- 
mines the  body's  place. 

Denote  its  co-ordi- 
nates by  x^  y^  z,  and 
those  of  the  element 
??2,  referred  to  this  point 
as  an  origin  by  a;',  y\ 
z' ;  there  will  thus  be 
two  sets  of  axes,  and 
sup})0sing  them  parallel, 
we  have 


and  differentiating, 


dx  =  dx^  -\-  dx\  ^ 
dy  —  dy,  -f  dy\  j> 
dz  =  dz,  +  dz\      ' 


(34). 


MECHANICS     OF     SOLIDS. 


59 


Demit  from  m,  the  per- 
pendiculars mX\  mY',  mZ' 
upon  the  movable  axes. 
Denote  the  first  by  ?',  the 
second  by  r",  and  the  third 
by  r"\  Let  0\  0",  0"\ 
be  the  projections  of  m^  on 
the  planes  x  y,  x  z,  y  2,  res- 
pectively. Join  the  several 
points  by  right  lines  as 
indicated  in  the  figure. 

Denote   the   ans;le 


Then   will 
the  triangle  m  Z'  0"  si 

tiie  triangle  in  Y'  0" 

(he  triangle  m  X'  0\ 


mZ'  0"  by  9, 
m  X'  0'  by  ^, 
m  Y'  0'"  by  4.. 


ive  J 


r-  s^ 


i 


X    =  r     cos  9, 


y'  =  ■;•'"  sin  9,  )_-  /^^,^  ^ 


(   x'  =  r"  sin  -v]^, 


■    sui  -^j  I 
"  cos  ^].,  j 


yl       _-      J./     ^Qg     ^^ 

z'  =  r'  sin  ■ra'. 


(35), 
(36), 
(37). 


We  here  have  two  values  of  x',  one  dependent  upon  9,  and  the 
other  upon  -^.  If  the  body  be  turned  through  an  indefuiitely  sniaJl 
angle  about  the  axis  s',  the  corresponding  increment  of  x'  is  obtained 
by    ditrereutiuling    the    /ii'st    of  Equations    (35)  ;    and    we    have 

d%'  ^  — r'"  sin  9  .  c^  9  ; 
if    it    be    turned    through    a    like    angle    about    the    axis    y',    tlie     cor- 
responding   increment    of  x'    is   found    by    differentiating    the    tiibi    of 
Equations  (3(3),  and 

d  x'  =  r"  cos  4'  •  (^  4'- 

[f    these    motions    take    place    simultaneously    about    both    axes,    the 
above    become    partial    ditferenlials    of   x\  and    we   iuivc   for   its   total 

differential, 

d  x'  ^^  r"  cos  -If,  .d\  —  v'"  sin  (p  .  d  <p, 


60 


ELEMENTS  OF  ANALYTICAL  MECHANICS. 


replacing  r"  cos  \  and  r'"  sin  (p,    by  their    values  iu  the  above  Equa- 
tions, and  we  get 

d  x'  =  z'  .d-\i  —  y'  .dQ^\ 
and  in  the  same  way, 

dy'  =  x' .  d^  —  z' .  d  -a, 
dz'  =  y'  .d^  —  x'  .d  4^,    J 

which  substituted  in  Equations  (34),   give 


} 


(38) 


dx  =  dx^  -\-  z' .  d  -^^  —  y'  .  d  (p,  '] 
dy  ^^  d y^  -\-  x'  .d  9  —  z'  .  f/w,  y 
dz  ^  d  z^  -\-  y' .  d  -ui  —  ,r '.  d  y.    I 


.(39) 


and  because  the  displacement  is   iiidefinitely   small,   Ave  may  write 

.     .     .(39)' 


8  x  =  5  x^  -\-  z'  .S-\j  —  y'  -^  ?, 

oy  =  Hy^  -\-  x' .  5  (p   —  z'  .S-u^, 
Sz   =  S z^  -\-  y'  .O-m  —  x'  .S-^; 


and  tnese  in  Equations  (31)  and  (32),  give 

P  cos  a .  ^.i-^  +  P  cos  [3 .  Sy^  +  P  cos  y .  Sz^ 

„    ^  -j-  J\  (.f' .  cos  (3  —  */' ,  cos  a) .  5xi 

P  .0  p  =    <  ^  ■  '     ^ 

+  /' .  {z'  .  cos  a  —  x' .  cos  y)  .  S-^ 
.  +  P'  .{y' .  cos  y  —  z' .  cos  /3) .  S-m. 


d^x     -       ,  d'^y     -.       ,  d^z 

7)1  •  -T-TT  •  ox^  4-  m  •  -— - .  5y^  +  m  ■ 


dl?     '"'    '    ■■'     dP 

x' .  d-y  —  y'  .  d'^x 


dfi 


^z. 


d,H  , 


+  m 


+  m 


dl- 

z' .  d~x  —  x' .  d'^z 
'  'dC- 


y' .  d"z  -  z'  .  d-^y    . 
dl- 


d-s 


Similar   values  may  be  found  for  /".Jw'and  7)1' . ——- •  ()s\  &:c.     in 

d(- 

these  values  ^.r^ ,  S y^     and  <^ Zj ,  will  be  the  same,  as  also  Scp^S-^,   and 

5^,   for   the   first  relate  to  ihe    movable    origin,  and  the  latter  to  the 

angular  rotation   which,  since  the  body   is   a  solid,  must  be  of  equal 


MECHA.iSriCS     OF     SOLIDS. 


61 


values  for  all  the  elements  ;  so  that  to  find  the  valnos  of  the  virtual 
moments  of  the  other  forces,  it  will  be  only  necessary  suitably  to 
accent  P,    a,  /3,  7,    x,  y,  2,    x\  y\  z'. 

These   values    being   found   and   substituted   in    Equation    (30),    we 
shall  find, 

(„  d"X\    ^ 

2  P .  cos  a  —  2  m  •  -— -  1  0  x, 
dp  /    -  *  ' 

+  (^2  P.  cos /3-  2;?«-^|-)  Sy,^ 

+   (2P.cosy-2,..i|-)o%., 

t^   ,   ,          ^        ,           X                -v' .  d"ii  —  11' .  d-x~\    ^ 
2  P.  (.!•'.  cos  /3— ?/'.  cos  a)  —2  »i .  •  ■" 0 

r      :„  ,  ,                  ,           ,               z' .  d-x  —  x' .  d'z  n   J.  , 
2P.  (2'.  cos  a— a;,  cos  7)— 2  m — o-j/ 


(  y  =0.(40) 


+ 


+       2P.  (?/'.  C0S7— s'.  cos/3)— 2 


y' .  d~z  —  s 


df 


s\d^l 


^^ 


But  the  displacement  being  entirely  arbitrary,  the  least  considera- 
tion will  show  that  Sx^,  8y^^  dz^,  Sep,  5  4/,  and  ^-ss'varc  wholly  inde- 
pendent of  each  other,  and  this  being  the  case,  the  principle  of  inde- 
terminate co-efficients  requires  tliai 

y^-7Jod  /  -         2  P .  cos  a  —  2  m  •  -^  =  0,    ^ 


M 


dC- 

dht 
2  P  .  cos  /3  —  2  m  .  —i-  —  0,)- 
dfl  '     ' 

d^-z 
2  P .  cos  7  —  2  »i .  --—  —  0 ; 
'  dt-  ' 


(-4) 


r  -  2  P.  {x' .  cos  ^  —  y' .  cosa)  —2m •'-^- =  0, 


dp 
z' .  d~x  —  x'.  d"Z 


'{  ~   2  P .  (z' .  cos  a  —  x' .  cos  7)  —  i  ... 

^  ''  dt^ 

V  ^    ,   ,  ,  o\        ^         y'-  c^"2  —  z'.  dhj 

A  =  2  P  .  (,y' .  cos  7  —  z' .  cos  /3)  _  2  -    ^  -^ 


rf/'-i 


=  0, 
-r  0. 


V.     .   (P) 


iiM>4U-'. 


r 


<u^  XT 


02  ELEMENTS     OF     ANALYTICAL     MECHANICS. 

§70. — These  six  equations  express  either  all  the  circumstances  of 
motion  attending  the  action  of  forces,  or  all  the  circumstances  of 
equilibrium  of  the  forces,  according  as  inertia  is  or  is  not  brought 
into  action ;  and  the  study  of  the  principles  of  Mechanics  is  little 
else  than  an  attentive  consideration  of  the  conclusions  which  follow 
fi'om  their  discussion. 

Equations  {A)  relate  to  a  motion  of  translation,  and  Equations 
(jB)  to  a  motion  of  rotation.  They  are  perfectly  symmetrical  and 
may  be  memorized  "with  great  ease. 

COMPOSITION    AKD    KESOLUTIOISr    OF    FOECES. 

^  §  77. — AVhen   a  free    body  is   subjected    to   the   simultaneous   action 

of  several  extraneous  forces  ■which  are  not  in  equilibrio,  its  state  ^vill 
be  changed ;  and  if  this  change  may  be  produced  by  the  action  of 
a  single  force,  this  force  is  called  the  resultant,  and  the  several  forces 
are  termed  comjjonents. 

The  resultant  of  several  forces  is  a  single  force  which,  acting  alone, 
will  produce  the  same  effect  as  the  several  forces  acting  sinutltancottslg ; 
and  the  comjionents  of  a  single  force,  ai'e  several  forces  whose  simvi'a- 
neous  action  2^'>'oduces    the   same   effect   as    the   single  force. 

If,  then,  several  extraneous  forces  applied  to  a  body,  be  not  in 
equilibrio,  but  have  a  resultant,  a  single  force,  equal  in  intensity  to 
this  resultant,  and  applied  so  as  to  be  immediately  opposed  to  it, 
■will  produce  an  equilibrium;  or,  what  amounts  to  the  same  thing, 
if  in  any  system  of  extraneous  forces  in  equilibrio,  the  resultant  of  ail 
the  forces  but  one  be  found,,  this  resultant  will  be  equal  in  intensity 
and  immediately  opposed  to  the  remaining  force  ;  otherwise  the  sys- 
tem could   not   be   in  ecpiilibrio. 

Conceive  a  system  of  extraneous  forces,  not  in  equilibrio,  and 
applied  to  a  solid  body,  and  suppose  that  the  equilibrium  may  be 
jjroduced  by  the  introduction  of  an  additional  extraneous  force. 
Denote  the  intensity  of  this  force  Ijy  R,  the  angles  -which  its  direc- 
tion  makes  with  the  axes  x,  y  and  z,  by  a,  h  and  c,  respectively, 
and  the  co-ordinates  of  its  point  of  application  by  x,  y,  z.  Then, 
because   the  inertia  cannot  act,  d'^x,  cPy,  d-z  will  be   zero,  and  taking 


MECHANICS    OF    SOLIDS.  63 

the   two  origins  to  coincide,  Equations  {A)  and  {B).  will  give 

^  cos  a  +  P'  cos  a'  +  P"  cos  a"  4-  F'"  cos  a'"  +  &c.  —  0. 
P  cos  6  +  i"  cos  3'  -4-  P"  cos  }S"  +  P'"  cos  /3'"  +  ifcc.  =  0, 

^  cos  c  +  P'  cc.s  >  '  —  P"  cos  7"  +  P'"  c-os  y"  +  ,kc.  =  0  ; 


B{zcosb  —  y  c-os  a)  +  P'  (z'  cos  jS'  —  /  cos  aQ 
+  P"  (x^'  cos  3"  -  y"  cos  a")  -f  &c. 

^  (2  C-05  a  —  X  cos  c)  -^  P'  {s^  cos  a'  —  ar'  cos  "/) 
-L  P"  (2"  c-os  a"  —  «"  c-os  7")  +  &c. 

P  (y  cos  c  —  z  cos  i)  -{-  P*  (y'  cos  7'  —  2'  cos  3') 
+  P"  {y"  cos  /'  -  2"  cos  8")  4-  ic. 


1-- 


Now  R  is  equal  in  intensity  to  the  resultant  of  all  the  other 
forces  of  the  system,  or  lq  other  words,  to  the  resultant  of  all  the 
original  forces ;  and  if  we  give  it  a  direction  directly  opposite  to 
that  in  which  it  is  supposed  to  ac-t  in  the  above  equations,  it  be- 
c-omes  in  all  respects  the  same  as  that  resultant,  being  equal  to  it 
in  intensity  and  having  the  same  point  of  application  and  line  of 
direction.  Adding,  therefore,  180°  to  each  of  the  angles  a,  6,  and  c, 
the  first  terms  of  the  foregoing  equations  become  negative,  and 
transposing  the  other  terms  to  the  second  member  and  fh^tnging  all 
the   signs,  we-  have, 

R  ros  a  =  P'  COS  a'  -f  P"  COS  a."  -f  P"  COS  a"  —  vice.  =  X:  ^,  .^^J>S^tt^ 

R  rry.  h  =  P'  COS  ,3'  +  P"  COS  /3"  +  P"'  COS  jS"'  -I-  vice.  =  Y;\..  (41)  ;^-^Tf^ 
R  COS  •  =  F  COS  7'  +  P"cos  7"  +  P'"  COS  7'"  4-  «icc.  =  Z.  J 

f     P'(x'cos^' —  y'cos  a')   1  ^ 

R  (X  cos  J  —  y  cos  a)  =  ^  -{-P"  (x"  c-os  3"  —  y"  cos  a")  V  =Z ; 

t+&c.  J 

r     P  (2'  cos  a'  -  a/  cos  70   ' 
^  (2  cos  a  —  j:  cos  c)  =  ^  +P"  (^"  cos  a"  -  x"  cos  /') 

r     P'  (y'  COS  /  -  2'  cos  /3') 
S  (y  cos  c  -  2  cos  i)  =  J  -f  P'  (y"  COS  7"  -  2"  cos  /3") 


M; 


=N. 


'  (42) 


Gi  ELEMENTS     OF     ANALYTICAL    MECHANICS. 

or, 

li  cos  a  =  X, 

Bcosb  :^  r,   }■ (43) 

R  cos  c  ^  Z. 

M  (.r  cos  b  —  y  cos  a)  =  L, 

B  [z  cos  a  —  .r  cos  c)  =  31,   ^'- (4-i) 

B  [y  cos  c  —  z  cos  i)  =  iV.  j  ^ 

/■    • 
Eliminating    B  cos  a,    i2  cos  h    and    i2  cos  c,    from    Equations    (44), 
Dy  means  of  Equations  (43),  we  get,  by  transposing  all  the  terms  to 
the  first  member,  X'y  z  -  Y  y.  z,  ^d.-L  -  o 

^y.~J>ii.  -h  -^ty^  0  Xy  -  Yx  +  Z  =  0,  V2pt^  -  /^z  +Ii.^Al<f  =  <y 

L-J^.^  ^"-^'-^  '  Zx-  Xz  +  M  =  0,   I  X-  r  ^.y  t  'f  '-.  ^(45) 

These  are  the  equations  of  a  right  line.  But  .-c,  y  and  z  are  the 
co-ordinates  of  the  point  of  application  of  the  resultant ;  they  are, 
therefore,  the  equations  of  the  line  of  direction  of  the  resultant  i?, 
and  hence  the  point  of  application  of  the  resultant  may  be  taken 
iiuywhcre  on  this  line  M'ithout  clianging  its  effect.  Any  condition, 
therefore,  expressive  of  the  simultaneous  existence  of  these  equations, 
will  also  express  the  existence  of  this  single  line,  and  of  a  single 
resultant   to    the    system  of  forces. 

§78. — To  find  this  condition,  nudliply  the  first  of  these  Equations 
by  Z,  the  second  by  F,  the  tliird  by  X,  and  add  the  products; 
we   obtain, 

ZL+YM^XN^.^ (4G). 

§79. — Having  ascertained,  by  the  verification  of  this  Equation, 
that  the  forces  have  a  single  resultant,  its  intensity,  direction,  and 
the  equations  of  its  direction  may  be  readily  found  from  Equations 
(43)  and  (44). 

Squaring   each  of  the   group  (43),  and   adding,  \ve   obtain, 

JS2  (cos2  a   +   C0S2  h   +   C0S2  c)    =   A^2   +    y2   4.   Z2. 


MECHAAMCS    OF    SOLIDS.  65 

Extracting   the    square   root    and   reducing   by  the   relation, 
cos^  a  +  cos^  h  +  cos^  c  :=  1, 


there  will  result, 


B  =  V.Y2  +  Z2  4_  2? 


(47) 


which   gives  the  intensity   of  the  resultant,    since   JT,    Y  and  Z  are 
knotvn. 


Again,  from  the  same  Equations, 

X     ^ 

Y 


cos  a  =  — , 
It 


cos  6  = 


cos  c  = 


(48) 


which  make  kno'wn   the   direction  of  the   resultant. 
The   group  of  Equations  (45)  give, 

Xy  —Yx  +  :^P'  (cos  /S'  x'  -  cos  a'  y')  =  0,  ] 
Za;-X2  +  2P'  (cos  a'  z'  —  cosy'  x')  -  0,  [    .     .     • 
Yz  ~  Zy  +^F'  (cos  7'  y'  -  cos  /3'  2')  =  0.  j 

which   are   the   equations  of  the    direction  of  the  resultant. 


(49) 


PAIlAiLELOGKAil   OF   FOECES. 

''  §  80. — If  all   the  forces  be   applied   to  the   same  point,  this  point 

may  be  taken  as   the   origin  of  co-ordinates,  in  which  case, 

x'  =  x"  =  x'"  &c.  =  0, 
y'  =.  y"  ^  y'"  &c.  =  0, 

z'  =  z"  =  z'"  &c.  =  0, 

and   the   last   term  in   each  of  Equations    (49),  "s\nll   reduce   to   zero. 
Hence,  to   determine   the   intensity,   direction   and   equations    of»-  the 


6G 


ELEMENTS     OF    ANALYTICAL    MECHANICS. 


line    of    direction    of    the   resultant,    we    have,    Equations    (47),    (48) 
and  (49), 


B  =  yX2  +  F2  +  Z2 


cos  0  =  —^ 
Jti 


(50) 


(51) 


Xy  -  Yx  =  0,^ 

Zx  -  Xz  z=o\ (52) 

Yz-  Zy  =  0.^ 

The  last  three  equations  show  that  the  direction  of  the  resultant 
passes  through  the  common  point  of  apj^lication  of  all  the  forces, 
which   might   have   been   anticipated. 

/Z         §S1. — Let  the  forces  be   now  reduced   to  two,  and  take  the  plane 
of  these  forces  as   that  of  a;  y  ;    then  will 

y'  =  j"  -  Y"  =  &c.  =  90°  ;  2  =  0, 

the   last   Equation  of  group  (41)  reduces   to, 

f  2  =  0; 

and   the   above  Equations  become, 


i2  =  V^M^~r^ 


X 

cos  a  =  — ) 


cos  h 


J. 


(53) 
(54) 


cos  c  =  0, 

Xy  -  r,r  =  0 (55) 

Tlic  last  is  an  equation  of  a  right  line  passing  through  the 
origin.  The  direction  of  the  resultant  will,  therefore,  pass  through  the 
point  of  application  of  the  forces.  The  cos  c  being  zero,  c  is  90°, 
and  the  direction  of  the  resultant  is  therefore  in  the  p)l<^ne  of  the  forces. 


MECHANICS     OF    SOLIDS. 


ei 


Substituting  in    Equation   (53),    for  X  and    F,    their    values    from 
Equations  (41),  we  obtain, 

B  =  -\/  [P'  cos  a'  +  F"  cos  a")3  +  {F'  cos  f3'  +  F"  cos  f3"Y  ; 
and  since 


cos^  a'   +  cos^  /3'    =  1, 
cos^  a"  +  cos^  (3"  =  1, 

this  reduces  to 


R  =  ^p'2  +  p"2   +  2  i^'  F"  (cos  cc'  cos  oc"  +  cos  /5'  cos  /S") ; 

denoting   the   angle    made   by    the    directions  of  the  forces    by  S,   wc 

have, 

cos  a'  cos  a"   +  cos  (3'  cos  (3"  =   cos  S  ; 


and  therefore, 


E  =  -v/P'2  +  P"2  +  2  P'  F"  cos  S 


(56) 


from  Avhich  we  conclude  that  the  intensity  of  the  resultant  is  equal 
to  that  diagonal  of  a  parallelogram  whose  adjacent  sides  represent  the 
directions  and  intensities  of  the  components^  which  passes  through  the 
jjoint  of  application. 

§  82. — Substituting  in  Equations  (54),  the  values  of  X  and  Z,  from 
Equations  (41),  we  have, 

B  cos  a  =  F'  cos  a'  +  F"  cos  a", 
B  cos  h  z^  F'  cos  /3'  +  F"  cos  /3", 
and  because 

a'    =  90°  -  /3', 
a"  =  90°  -  /3", 
a     =  90°  —  5, 

these   Equations   reduce   to, 

i2  cos  a  =  F'  cos  a'  +  P''  cos  a", 
i2  sin  a  =  F'  sin  a'  +  P"  sin  a" ; 


iw.^gf^^''ELE]yrE'irfs  'of    ANiXTrl'^AL    MECHANICS. 

by  transposing    and    squaring,  we    obtain, 

P"2  (?os2  a"  =  i22  cos^  a  —  '2  EF'  cos  a  cos  a'  +  P'2  cos2  a', 
P''2  sin2  a"  =  i22  siii2  a  —  2  Ji  P'  sin  a  sin  a'  +  P'2  sin2  a' ; 

adding    and    reducing, 

P"2  ^  B"  +  P'^  -  2EP'  cos  (a  -  a')  ; 
but, 

a  —  a'  =  the  angle  i?  »i  P'  nr  9' ; 

hence,  by  transposition  and  reduction, 

i?2    _|_    p'3  _  p''2 

"^'  'p  =  — 2:^p^ — ' 

or, 

P"2_fjl_P'y       (P"J^R-P')(P"  +  P'-R) 

l  —  ooso)'— 2sin2l-a)  — '^ —  —  -^ ■ — ■ ; 

vdience,  making 

EJrP'+P"         „ 

2 ""      ' 

vre    obtain. 


s"ii9'-\/ ^^r (50 

from  -which  we  see  that  the  direction  of  the  resultant  coincides  with 
the  diagonal  of  the  parallelogram  described  on  the  lines  represent- 
ing   the    intensities   and   directions  of  the   forces. 

Thus,  the  resultant  of  any  two  forces,  upiylied  to  the  same  material 
point,  is  re2}rese)ited,  in  intensity  and  direction,  by  that  diagonal  of  a 
parallelogram,  constructed  tqmn  the  sides  rejyresenting  the  intensities 
and  directions  of  the  tivo  components,  which  j^u^^'cs  throuyio  the  point 
of  apjilicatio)!. 

§83. — In  the  triangle  RmP',  since  P'  R  is  equal  and  parallel  to 
the  line  which  represents  the  force  P",  the  angle  niF'R  =  9,  is  the 
supplement  of  the  angle  S,  made  by  the  directions  of  the  components, 
and  there  will  result  the  following  equation^:    '/^—     /S^o—      ■    x^— ^^ 

1  .         -1             7(6'-  I")    (S  -  P")  ._. 

cos  1  5  z:^  sm  \(p  =  \J  ^ -jjrjrr, ;  '  *  (5») 


.   ^      r     • 


MECHANICS     OF    SOLIDS, 


fiO 


Equation  (57),  will  make  known  the  angle  made  by  the  direction 
of  the  resultant  with  that  of  either  of  two  oblique  components,  pro 
vided,  the  intensities  of  the  components  and  resultant  be  known. 

§84. — Also,  from  the  two  triangles  RmF'   and    RmP'\    we  find,  ,    ^ 


sn:  9    = 


sni  cp 


F" 

.  SUl  0 

R 

E' 

.  sin  S 

>.  ■  ■  (69)?2 


'"^^'--•^ 


R 


J 


from  which  the  angles  made 
by  the  direction  of  the  result- 
ant with  its  two  components  may  be  found. 

§  85. — Let  there  now  be  the  three    forces    P,   P',   P'\    applied    to 
the     material     point     ?h,     in 
the     directions    vi  P,    m  P\ 
m  P",  not  in  the  same  plane ; 
the  resultant   will  be   repre- 
sented in  intensity  and  direc- 
tion  by    the   diagonal   of    a 
parallelopipedon,  constructed 
upon   the   lines   representing 
the  directions  and  intensities 
of  these   components.      For, 
lay    off   the    distances    iu  A^ 
m  C\  and    m  E^   proportional 
to  the  intensities  of  the  com- 
ponents  which   act   in   the   direction  of   these  lines,  and  construct  the 
parallelopipedon    E  B  ;    the    resultant    of    the    components   P'   and    P 
will,    §82,    be   represented  by  the  diagonal  m  B^  of  the  parallelogram 
mAB  C  ;     and  the  resultant  of  this  resultant  and  the  remaining  com- 
ponent P",  will  be  represented  by  the  diagonal  in  D  of  the  parallelo- 
gram   EmBD.    which   is   that  of  the  parallelopipedon. 

§  86. — If  the  forces  act  at  right  angles  to  each  other,  the  parallel- 
opipedon will  become  rectangular,  and  the  intensity  of  the  resultant, 
denoted   by  P,  will   become  known  from  the  formula 


ru 


ELEMENTS     OF    ANALYTICAL     MECHANICS. 


R  =   -/P2  4.  pn  ^p"2  . 


and  if  the  angles  which  the 
direction  of  the  resultant 
makes  with  those  of  the 
forces  P,  P'  and  P",  be 
represented  by  a,  b,  and  c, 
respectively,  then  will 

R  cos  a  ^^  P^ 
Rcosb  z=  P', 
Rcosc  =  P". 


^ .-rA  ^Vree  lines  be  drawn  through  the  point  of  application   m',   of 
t,  .   fnir^-e  P\  parallel  to  any  three  rectangular  axes  x,y,z',  and  denote 
by  ■)i',  p\  J  \  the  angles  which 
th ;    direction    of    this    force  ^ 

iruikes   with   th^^e   axes   res- 
pectively ;    then  -vv'ill 


P'  cos  a', 
P'  cos  /S', 
P'  cos  7', 


be  the  components  of  the  force  P',  in  the    direction  of  the  axes,  and 

they  will  act  jilong  the  linec  drawn  through  the  point  m'.      These  are 

the  same  as  the  ieims   composhig   in   part  Equations  (A),  and  as  the 

effect  of  the  conipv^iVP.ts  is  identical   with  that  of  the   resultant,  tliese 

components    may  a.'wais   be   substilisted  for  the  foj-ce  P'.      I'he  same 

dP'X        (P'u  cl^s 

for    the     forcfs    of   inef^M,   and    in--—-j   in-—^i   and  m — -)    denote    tht 

coniponenls    ol    liiis    J'ore^'.    in    the    direcLioiis    uf    the    axes. 


MECHANICS    OF    SOLIDS. 


71 


^ 


§87. — Examples. — 1.  Let  the  point  m,  be  solicited  by  two  forces 
whose  intensities  are  9  and  5,  and  whose  directions 
malce  an  angle  with  each  other  of  57°  30'.  Re- 
quired the  intensity  of  the  force  by  which  the 
point  is  urged,  and  the  direction  in  which  it  is 
compelled  to  move. 

First,  the  intensity  ;    make  in  Equation  (56), 

P'   =  9, 
F"  =  5, 
S  =  57°  30'  ; 

and  there  will  result, 

B  =  V  81  +  25  +  90  X  0,  537  =  12,422. 

Again,  substituting   the   values  of  5,  F'  F"  and  R  in  the  first  of 
Equations  (59),  we  have, 

5  X  sin  57°  30' 


sm  9    1= 


12  122 


or, 


9'  =  19°  50'  35"  nearly, 


which  is  the  angle  made  by  the  direction  of  the  force  9  with  that  of 
the  resultant. 

2. — Required  the  angle  under  which  two  equal  components  should 
act,  in  order  that  their  resultant  shall  be  the  «'*  part  of  either  of  them 
separately. 

By  condition,  we  have 


hence. 


F'  +  F"  +  R 


S 


F"  =  nR 


iR  +  nR  +  R  _  (2/1  +  \)  R 


and,  Equation  (58), 

sin  ^  (p 


=v 


{S  -  P')  {S  -  F") 

p,  pn 


72  ELEMENTS     OF    ANALYTICAL     MECHANICS. 

which  reduces  to 

If  n  be  equal   to   unity,  or  the  resultant  be  equal  to  either  force, 

9  =  00°, 
and,  §83,  the  angle  of  the  components  should  be  120°. 

3. — Required  to  resolve  the  force  18  =  a,  into  two  component? 
whose  difference  shall  be  5  =  &,  and  whose  directions  make  with 
each  other  an  angle  of  38°  =  §.  Also,  to  find  the  angle  which  the 
direction  of  each  component  makes  with  that  of  the  resultant. 

Writing  a  for  JR   in  Equation  (56),  we  have, 

P'2  ^  p//2  +  2  P'  F"  cos  S  =  a?, 
and  by  condition. 

P'  -  P"  ^b (  c ). 

Squaring  the  second  and  subtracting  it   from  the  first,   we  get 

^P'P"  (1  +  cos  (J)  =  a2  _  ^2  . 
which,  replacing  (1  +  cos  8)  by  2  cos^  |-  (J,  reduces  to 

cfl  —  U^ 

4P'  P"    zzz . 

cos^  \  6 
This  added  to  the  square  of  the  Equation  ( c ),  gives 


from  which  and  Equation   ( c )  we  finally   obtain. 


^'  cos^  i-  0 


=^-(-x/^^^=^SiF2'+0--.o«. 


which  are  the  required  components. 

To  find  the  angles  which  their  directions  make  with  the  resultant, 
we  have  from  Equations  (59), 

cp".  =  24°  —  the  angle  which  F"  makes   with  the  resultant. 


-9 

MECHANICS     OF    SOLIDS. 


n 


and, 


9' 


14° 


ano-le  which  P'  makes  with  the  resultant. 


4. — Eequired  the  angle  under  which  two  components  whoso  inten- 
sities are  denoted  by  5   and  7  should   act,  to  give  a  resultant  whose 

intensity  is  represented  by  9. 

An?.  84°    15'    39"    ■ 

5. — From  Equation  (50)  it  appears  that  the  resultant  of  two 
components  applied  to  the  same  point,  is  greatest  when  the  angle 
made  by  their  directions  is  0°,  and  least  when  180°.  Eequired  the 
angle  under  which  the  components  should  act,  in  order  that  the. 
resultant  aiiay  be  a  mean  proportional  between  these  values;  and 
also  the  angle  which  the  resultant  makes  with  the  greater  component 
Call  P',  the  greater  component. 

'    ',.  1 1  Ans.  S 

6. — Given  a  force  whose  intensity  is  denoted  by  17.  Required  the 
two   comf>onents   which  make  with  it  angles  of  27°  and  43°.  "^jC 

§  88. — The  theorem  of  the  parallelogram  of  forces,  just  explained, 
enables  us  to  determine  by  an  easy  graphical  construction  the  in- 
tensity and  direction  of  the  resultant  of  several  forces  applied  to  the 
same   point. 

Let  P\  P'\  P"\  &c.,  be 
several  forces  applied  to  the 
same  point  m.  Upon  the 
directions  of  the  forces,  lay 
off  from  the  point  of  ap- 
plication distances  projior- 
tional  to  the  intensities  of 
the  forces,  and  let  these  dis- 
tances represent  the  forces. 
From  the  extremity  P'  of 
the   line    m.  P\    which    repre- 


JP' 


i^iy 


n-^— 


74: 


ELEMENTS     OF     ANALYTICAL    MECHANICS. 


sents  the  first  force,  draw  the  line  P'  n  equal  and  parallel  to  m  P" 
which  represents  the  second,  then  will  the  line  joining  the  extremity 
of  this  line  and  the  point  of  application,  represent  the  resultant  of 
these  two  forces.  From  the  extremity  w,  draw  the  line  n  n'  equal 
and  parallel  to  mP'"  which  represents  the  third  force;  m  n'  will 
represent  the  resultant  of  the  first  three  forces.  The  construction 
being  thus  continued  till  a  line  be  drawn  equal  and  parallel  to 
every  line  representing  a  force  of  the  system,  the  resultant  of  the 
whole  will  be  represented  by  the  line,  (in  this  instance  m  n"),  join- 
ing the  point  of  application  with  the  last  extremity  of  the  last 
line  drawn.  Should  the  line  which  is  drawn  equal  and  parallel  to 
that  which  represents  the  last  force,  terminate  in  the  point  of  appli- 
cation, the   resultant  will    be    equal    to    zero. 

The  reason  for  this  construction  is  too  obvious  to  need  expla- 
nation. 

§  89. — If  the  forces  still  be  supposed  to  act  in  the  same  plane, 
but  upon  different  points  of  the  plane,  the  first  of  Equations  (49) 
takes    the   form, 

Yx  -  Xy  =  2  [P'  (cos  /3'  x'  -  cos  a'  y')  ], 
thus,  differing  from  Equation  (55),  in  giving  the  equation  of  the  line 
of  direction    of    the    resultant    an    independent     term,     and    showing 
that   this  line  no   longer  passes   through  the  origin.     It  may  be  con- 
structed from    the  above   equation. 

§  90. — To  find   the  resultant  in  this  case,  by  a  graphical  construc- 
tion, let  the  forces  P', 
P",  P'"    &c.,    be    ap- 
plied to  the  points  m', 
m'\   m"\  &c.,    respec-  ^^-a? 

lively.  Produce  the 
directions  of  the  fiirccs 
P'   and    P"   till    they 

meet   at    0,   and    take         /  •^' 

this    as   their  common      ^" 
point    of   application  ; 
lay  off   from    0,    on  ilie     ines   of  direction,  distances  OS  and  0  S\ 


J" 


0" jniv 


J2/ 


or 

"7\ 


\     / 


MECHANICS     OF     SOLIDS.  75 

proportional  to  the  intensities  of  the  forces  P'  and  P",  and  construct 
the  parallelogram  0  S  B  S',  then  will  0  M  represent  the  resultant  of 
these  forces.  The  direction  of  this  resultant  being  produced  till  it 
meet  the  direction  of  the  force  P"\  produced,  a  similar  construction 
will  give  the  resultant  of  the  first  resultant  and  the  force  P'", 
which  will  be  the  resultant  of  the  three  forces  P\  P'  and  P'"  \ 
and   the  same  for    the    other  forces. 

OF   PARALLEL   FORCES. 

§91. — If  the  forces  act  in   parallel    directions, 

cos  a'  =  cos  a."  =  cos  uJ"  =r   <fcc., 

cos,S' =  cos/3"  =  cos/3'"  =  &c., 
cos  y'  =  cos  j"  =  cos  y'"  =   &c., 

and  Equations  (41)  become, 

X  =  {P'  +  P"  +  P'"  +  &c.)  cos  a', 
r={P'  +  P"  +  P'"  +  &c.)  cos  /3', 
Z  =  {P'  +  P"  +•  P'"  +  &c.)  cos  y' ; 

these  values  in  Equation  (47)  give, 

R  =  ±      -^{P'  +  P"  +  P'"  +  &c.)2  (cos2  a'  +  cos2  /3'  +  cos2  P), 

but, 

cos2  a.'  -\-  cos^  /3'  +  cos^  y'  —  1  ; 
hence, 

R  ^  P'  +  P"  +  P'"  +  &c. (GO) 

If  some  of  the  forces  as  P",  P'",  act  in  directions  opposite  to 
the  others,  the  cosines  of  a"  and  aJ"  will  be  negative  while  they 
have    the    same   numerical   value ;  and   the   last  equation  will  become 

R  =  P'  -  P"  -  P'"  +  &c. 

Whence  we  conclude,  that  the  resultant  of  a  number  of  parallel 
forces  is  equal  in  intensity  to  the  excess  of  the  s^tm  of  the  inte.i- 
sifies  of  those  which  act  in  one  direction  over  the  sum  of  tht 
intensities    of  those    lohich    act    in    the    opposite  direction. 


■76  ELEMENTS     OF    ANALYTICAL    MECHANICS. 

§  92. — The  values  of  H,  X,   V  and    Z   being    substituted    in  Equa- 
tions  (48)  give, 

(P'  4-  P"  +  P'"  +  &c.)  cos  a'  _ 


cos  b  =  -  ,^,    , — r-y— — ^^,,,    ,     „ — ' =  cos  fd', 


F'  +  P" 

+  P'"  +  &c. 

{P'  +  P" 

+  P'"  +  &c.)  cos  /3' 

P'  +  P" 

+  P'"  +  &c. 

{P'  +  P" 

+  P'"  +  &c.)  cos  y' 

P'  +  P"  +  P'"  +  &c.  ~  ^^'^  ^  • 

The  denominator  of  these  expressions,  being  the  resultant,  is  essen 
tially  positive ;  the  signs  of  the  cosines  of  the  angles  a,  h  and  c. 
will,  therefore,  depend  upon  the  numerators ;  these  are  sums  of  the 
components  parallel  to  the  three  axes. 

Hence,  the  resultant  acts  in  the  direction  of  those  forces  lohose 
cosine  coefficients  are  negative  or  positive  according  as  the  sum  of  the 
former  or  latter  forces  is  the  greater. 

§03.— The  forces  being    still    parallel,  Equations  (42)  reduce  to, 

(        (P'  *■'  +  P"  x"  +  P'"  x'"  +  &c.)  cos  /3 
R  X  cos  0  —  Ry  cos  a  =  ^ 

\  -  {P'f  +  P"y"  +  P'"  y'"  +  &c.)  cos  a 

(P'  z'  +  P"  z"  +  P'"  z'"  +  &c.)  cos  K 
R  z  cos  a  —  Rx  cos  c  =  •< 

I  -  {P'x'  +  P"  ;f"  +  P'"  x'"  +  &c.)  cos  7 

J       (P'y'  +  P"/'  +  P"'y'"  +  &c.)cos7 
ic  y  cos  c  —  B  z  cos  o  =  ^ 

-^  (   -  (P'  .?'  +  P"  s"  +  P'"  z'"  +   &C.)  CO:,  p 

but, 

COS  6  =  COS  /3', 

COS  a  r=  COS  a', 

COS  C  :=  COS  7' ; 

Substituting  the  second  members  of  these  last  equations  in  the 
first  of  the  equations  immediately  preceding,  and  ti-ansposing  all  the 
tei'uis    to  the  first  member,  we  obtain. 


[Rx  -  {P'x'  +  P"x"  H-  P"'.x"'  +  &c.)]  cos/3' 

-  [Rg  -  {P'y'  4-  P"?/"  +  P"'y'"  4-  &c.)]  cos  a' 

\Rz  -  (P'  2'  +  P"  z"  +  P'"  £•'"  4-  &c.)]  cos  a'  I  _ 

-  \Rx  -  (7^'.r'  +  P"  x"  +  P"'.c'"  +  &c.)j  cos  7M'  "^    ' 

[Py  -  {P'y'  +  P"y"  +  P'" y'"  +  &c.)]  cosy' 

-  [P  2  -  (P'  2'  +  P"  2"  +  P'"  z'"  +  &c.)]  cos/3 


[=0. 


,'[=0. 


MEOHANICS    OF    SOLIDS. 


77 


These  equations  must  be  satisfied,  whatever  may  be  the  angles 
wliioh  the  coininon  direction  of  the  forces  malces  with  the  co-ordinate 
axes,  and  this  can  only  be  done  by  making  the  co-efficients  of  the 
cos  a',  cos/o"  and  cosy',  (either  two  of  the  latter  being  arbitrary), 
separately  equal    to   zero.     Hence, 


Ex  =  F'x'  +  F'\c"  +  F"'x"'  +  &c. 
El/  =  P',y  +  P"y"  +  P"'y"'  +  &c. 
Ez  =z  P'z'  +  P"z"  +  P"'z"'  +  &c.  j 


(61) 


The  forces  being  given,  the  value  of  i?,  §91,  becomes  known, 
and  the  co-ordinates  x,  y,  2,  are  determined  from  the  above  equations  ; 
these  co-ordinates  will  obviously  remain  the  same  whatever  direction 
be  given  to  the  forces,  provided,  they  remain  parallel  and  retain  the 
same  intensity  and  points  of  application,  these  latter  elements  being 
the  only  ones  ujdou  which  the  values  of  .r,  y,  2,  depend. 

The  point  whose  co-oi'dinates  are  x^  ?/,  2,  which  is  the  point  of 
application  of  the  resultant,  is  called  the  centre  of  imrallel  forces^  and 
may  be  defined  to  be,  that  point  in  a  system  of  j^arallel  forces  through 
u'hich  the  resultant  of  the  system  ivill  always  ^j«ss,  whatever  he  the 
direction  of  the  forces,  provided,  their  intensities  and  points  of  appli- 
cation remain   the  same. 


§  94. — Dividing  each  of  the  above  Equations  by  E,  we  shall  have 
P'x'  +  P"x"  -f  P"'x"'  -f  &c. 


y 


pr     ^    p„   j^   pn,   ^    ^^^ 

P'y'    +    P"y'>    J^    p>"y,n    _^     ^^^ 

~P'   -f   P"   +    P'"   +   &C. 

P'z'  +  P"z"  -f  P"'z"'  +  (fee. 
P'  +  P"  +  P'"  +  &c. 


((3?) 


Hence,  either  co-ordinate  of  iJie  centre  of  a  system  of  parallel  forces 
is  equal  to  the  algebraic  sum  of  the  products  which  result  from  multi- 
Inlying  the  intensity  of  each  force  by  the  corresponding  co-ordinate  of  its 
point  of  application,  divided  by  the  algebraic  sum  of  the  forces. 

If  the   points  of  application  of  the    forces    be    in    the    same    i  !aiie, 


TS 


ELEMENTS     OF     AlVALYTICAL    MECHANICo. 


the   co-ordinate   i^lane   .-c  y,    may   be   taken   parallel    to    this   plane,    in 
will  oh  case 


&:c. 


and. 


_    (P^  +  P"  +  P'"  +  &c.)  z'   _    ^ 
^  ~       P'  +  P"  +  P'"  +  &c.       ~  ^  ' 

from  which  it  follows  that  the  centre  of  parallel  forces  is  also  in  this 
plane. 

If  the  points  of  application  be  upon  the  same  straight  line,  take 
the  axis  of  x  parallel  to  this  line  ;  then  in  addition  to  the  above  results 
we  have 

y'  =y"  =  y'"  =  &zc.; 
and, 

(P'  +  P"  +  P'"  +  &c.)  y' 


y  = 


y 


P'  +  P"  +  P"'  +  &c. 

whence,  the  centre  of  parallel  forces  is  also  upon  this  line. 

§  95. — if  we  suppose  the  parallel  forces  to  be  reduced  to  two,  viz, 
P'  and  P",  we  msij  assume  the  axis  x  to  pass  through  their  points 
of  application,  and  the  plane  xy  to  contain  their  directions,  in  which 
case,  Equations  (60)  and  (01)  become, 

R  =  P'  +  P" 
Rx  =  P'x'  +  P"x" 
g  =  0   and   ?/  =  0. 


^Nfultiplying  the  first  by  x\  and  subtracting 
the  product  from  the  second,  we  obtain 

R  {x  -  x')  =  P"  (x"  -  x')  .  .  («) 

Multiplying  the  first  by  x"  and  sub- 
tracting the  second  from  the  product, 
we  get 


A      ii"     ^r 


R  {x"  -  x)  =  F'  [x"  -  x')     . 


(S) 


Denoting  by  ;S"  and  S",  the  distances  from  the  points  of  application 


MECHANICS     OF    SOLIDS. 


79 


of  P'  and  P"  to  that  of  the  resultant,  wliich  are  x  ~  x'  and  x"  -  -  x 
respectively,  we  have 

x"  -  x'  =  S'  +  S" ; 

and  from  Equations  (a)  and  (h),  tliere  will  result 

P'  :  P"  :  B  ::  S"  :   S'  :  S"  +  S'      .     .     .     .    (63) 

If  the   forces   act   in    opposite   directions,  then,   on   the    supposition 
that  P'  is  the  greater,  will 

B  =  P'  -  P" 
Bx  =  P'x'  -  P"x" 
z  =  Q,   y  =  0. 

and  by    a   process    plainly   indicated    by 
what  precedes, 


P'  -.P"  :B::  S"  :  S'  :  S"  -  S\ 


(64). 


From  tills  and  Proportion  (63),  it  is 
obvious  that  the  point  of  application  of 
the  resultant  is  always  nearer  that  of  the 
greater  component;    and    that    when    the 

components  act  in  the  same  direction,  the  distance  between  the  j)oint 
of  application  of  the  smaller  component  and  that  of  the  resultant,  is 
less  than  the  distance  between  the  points  of  application  of  the  com- 
ponents, while  the  reverse  is  the  case  when  the  components  act  in 
opposite  directions.  In  the  first  case,  then,  the  resultant  is  between 
the  components,  and  in  the  second,  the  larger  component  is  always 
between  the  smaller  component  and  the  resultant. 

And  we  conclude,  generally,  thai  the  resultant  of  two  forces  which 
r,Qlicit  two  points  of  a  right  line  in  jjaraJlel  directions,  is  equal  in  intcn- 
sidj  to  the  sum  or  difference  of  the  intensities  of  the  components,  accord- 
ing as  they  act  in  the  same  or  opposite  directions,  that  it  ahoays  acts 
in  the  direction  of  the  greater  comiJonent,  that  its  line  of  direction  is 
contained  in  the  jilane  of  the  comp>oncnts,  and  that  the  intensity  of  either 
component  is  to  that  of  the  resultant,  as  the  distance  between  the  point 
of  a2')pUcation  of  the  other  component  and  that  of  the  resultant,  is  to 
the  distance  between  the  points  of  apjilication   of  the  components. 


80 


ELEMENTS     OF     ANALYTICAL    MECHANICS. 


H 


.P" 


J". 


tiif 


§  9G. — Examples. —  1.  The  length  of  the  line  to' wi" "  joining  the 
points  of  application  of  two  parallel  forces 
acting  in  the  same  direction,  is  30  feet ;  the 
forces  are  represented  by  the  numbers  15 
and  5.  Required  the  intensity  of  the  re- 
sultant, and  its  point  of  application. 

i2  =  P'  +  P"  =  15  +  5  =  20  ; 

R    :  P'  ::  m"  m'  :  m"  o, 

20  :   15  ::  30  :  m"  o  =  23,5  feet. 

A  single  force,  therefore,  whose  intensity  is  represented  by  20,  applied 
at  a  distance  from  the  point  of  application  of  the  smaller  force  equal 
to  22,5  feet,  will  produce  the  same  effect  as  the  given  forces  applied 
at  m"  and  m' . 

2. — Required  the  intensity  and  p<Vint 
of  application  of  the  resultant  of  two 
parallel  forces,  whose  intensities  are  de- 
noted by  the  numbers  11  and  3,  and 
which  solicit  the  extremities  of  a  right 
line  wliose  length  is  16  feet  in  opposite 
directions. 


R  =  P'  —  P"  ^  11 
P'  -  P"  :  P'  ::   m' 


P'  .  m"  on' 


=  22  feet. 


3. — Given  the  length  of  a  line  whose  extremities  are  solicited  in 
the  same  direction  by  two  forces,  the  intensities  of  which  differ  by 
the  n""  part  of  that  of  the  smaller.  Required  the  distance  of  the 
point  of  aj^plication  of  the  resultant  from  the  middle  of  the  line 
Let  2  Z,  denote  the  length  of  the  line.     Then,  by  the  conditions, 


P'    =z    P" 


n  \     n      y 

^      n      /  n 

\     n     y 


^ 


21  :  m'o  = 


!>• 


2)1  4-  1 


CO  =  I  — 


2nl 


1 


2n  +1       2/i  +  1 


:l. 


MECHANICS     OF    SOLIDS, 


81 


§97. — The  rule  at  the  close  of  §95,  enables  us  to  determine  by  a 
very  easy  graphical  construction,  the  position  and  point  of  application 
of  the  resultant  of  a  number  of  parallel  forces,  whose  directions, 
intensities,  and  points  of  application  are  given. 

Let  P,  P',  P'\  F",  and  P'\ 
l)e  several  forces  applied  to  the 
material  points  m,  m',  m",  m'", 
and  ??i"',  in  parallel  directions. 
Join  the  points  m  and  m'  by  a 
straight  line,  and  divide  this  line 
at  the  point  o,  in  the  inverse 
ratio  of  the  intensities  of  the 
forces  P  and  P' ;  join  the  points 
0  and  m"  by  the  straight  line 
0  m" ^  and    divide    this    line  at  o', 

in  the  inverse  ratio  of  the  sum  of  the  first  two  forces  and  the  force 
P"  \  and  continue  this  construction  till  the  last  point  m'"  is  included, 
then  will  the  last  point  of  division  be  the  point  of  application  of  the 
resultant,  through  which  its  direction  may  be  draAvn  parallel  to  that 
of  the  forces.  The  intensity  of  the  resultant  will  be  equal  to  the 
algebraic  sum  of  the  intensities  of  the  forces. 

The  position  of  the  point  o  will  result  from  the  proportion 


P  +  P'  :  P' 


P' 


P  +  P' 


that  of  o'  from 


P  +  P'  +  P"   :   P"  :  :  o  m"  :  o  o' 


P" .  0  m 


P  +  P'  +  P' 


that  of  o"  from 

P  +  P'  -f  P"  -  P'"  :  ~  P'"  :  o'  m'"  :  o'  o"  = 

and  finally,  that  of  o'"  from 


P'"  .  o'  m"' 


P+P'+P"-P" 


P+P'  +  P"-P"'-\-P''  :  P" 


P-\-P'+P"-P"'  +  P" 


1 


82        .     ELEMENTS     OF     ANALYTICAL    MECHANICS. 


OF    COUPLES. 

§98. — When  two  forces  P'  and  P"   act  in  opposite  directions,  the 
distance  of  the  point  o,  at  which  the  resultant 
is    ajiplied,  from    the    point   ?«',    at   which   the 
component   P'   is    applied,   is   found   from    the 

foimula 

J? 

/ 


m"  m' .  P" 


P' 


Ji 


and    if    the    components   P'    and   P"   become 

equal,  the    distance   m'  o   will   be   infinite,   and 

the  resultant,  zero.     In  other  words,  the  forces 

will    have    no    resultant,  and   their  joint  effect 

will  be  to  turn  the  line    iii"  vi\   about  some  point  between  the  points 

of  application. 

The  forces  in  this  case  act  in  opposite  directions,  are  equal,  but 
not  immediately  opposed.  To  such  forces  the  term  couple  is  applied, 
A  couple  having  no  single  resultant,  their  action  cannot  be  compared 
to  that  of  a  single  force. 

§  99. — The  analytical  condition,  Equation  (46),  expressive  of  the 
existence  of  a  single  resultant  in  any  system  of  forces,  will  obviously 
be  fulfilled,  when 

X  =  0,     Y  =  0,    and  Z  =  0. 

But  this  may  arise  from  the  parallel  groups  of  forces  whose  sums 
are  denoted  by  X,  P",  and  Z,  reducing  each  to  a  couple.  These  three 
couples  may  easily  be  reduced  by  comjjositiou  to  a  single  couple, 
bc}"ond  which,  no  further  reduction  can  be  made.  It  is,  therefore,  a 
failing  case  of  the  general  analytical  condition  referred  to. 

"VVOliK   OF   THE   KESULTANT   AND    OF    ITS    COMPONENTS. 

§  100. — We  have  seen  that  when  the  resultant  of  several  forces 
is  introduced  as  an  additional  force  with  its  direction  reversed,  it 
will   hold   its    components    in    equilibrio.      Denoting   the   intensity    of 


MECHANICS    OF    SOLIDS  83 

the    resultant    by    i?,    and   the   projection   of    its    virtual    velocity    by 
6  r,  we   have   from    Equation  (29), 

-  B5r  -{-  P.Sj)  +  P'.Sp'  +  P".5y  +  &c.  =  0, 


or. 


ESr  =  F.Sj^  +  F'  S/  +  P"  52J"  +  &o.,  ....     (65) 

in   which  P,  P'  P",  &c.  are    the    components,  and  S  p,  S p'  Bp)".  &c. 
the   projections  of  their  virtual  velocities. 

§  101. — Now,  the  displacement  by  which  Equation  (29)  was  de- 
duced, was  entirely  arbitrary  ;  it  may,  therefore,  be  made  to  conform 
in  all  respects  to  that  which  would  be  produced  by  the  components 
P,  P',  &c.,  acting  without  the  opposition  of  the  force  equal  and 
contrary  to  their  resultant;  and  writing  dr  for  (Jr,  dp  for  (Jj9,  &c., 
Equation  (65)  will   become 

Rdr  z=z  Pdp  -\-  P' dp'  +  P" dp"  +  &c.,  •     .     •     (66) 

and    integrating, 

JRdr  =  JPdp  -f  JP'dp'  +  fP"  dp"  +  &c.,  .     .     (67) 

in  which  P,  P,  P',  &c.  may  be  constant  or  functions  of  r,  p^  p',  &c., 
respectively. 

From  Equations  (66)  and  (07),  it  appears  that  the  quantity  of 
work  of  the  resultant  of  several  forces  is  equal  to  the  algebraic  sum 
of  the  quantities  of  work  of  its  components. 

x\gain,  replacing  P(5jj»,  P'  6 p\  &c.  in  Equation  (65),  by  their  values 
in    Equation  (-31),  and  writing  dr  for  (5  r,  dp  for  5 p^  &c.,  we  find, 

fRdr  =  flP.cosa.dx  +  fZP.cos  fS  .dy  -\-  f:zP.cos  y.dz,  ■  ■  (6>) 

ir  which  P    may  be   constant   or   a  function  of  r ;  P,  constant  or   a 
function    of  .r,  y,  0,  &c. 

If  the  forces  be   in    equilibrio,  then  will  E  =  0.  and, 

2 P.  cos-  a.dx  +  2 P.  cos  fS-d^j  +  SP.  cos  y.d z  —  0.  .     •     (69) 


S-i  ELEMENTS     OF    ANALYTICAL    MECHANICS. 


MOMENTS. 

§  102. — It  is  now  apparent  that  in  the  transformation  of  Equation 
(30)  to  Equation  (40),  each  force  of  the  original  system  was  rephiced 
by  its  three  components  in  directions  of  three  rectangular  axes,  arbitra- 
rily assumed. 

The  components  parallel  to  either  axis  will,  §  43,  work  duiing  any 
motion  which  will  carry  their  points  of  application  in  the  direction 
of  that  axis,  and  will  cease  to  work  when  the  motion  becomes  per- 
pendicular to  the  same  line. 

Let  the  points  of  application  of  tlie  forces  move  in  linos  parallel 
to  the  axis  a;;  the  components  paiallel  to  z  alone  can  work,  for  the 
23aths  being  perpendicular  to  the  directions  of  the  other  components, 
the  v\-ork  of  the  latter  will  be  nothing,  because  the  projections  of 
the  paths  upon  their  lines  of  direction  will  be  zero.  The  elementary 
v.ork  of  the  extraneous  forces  will,  in  this  case,  be  found  in  the  third 
term  of  Equation  (40),  and  equal  to 

(2  P  cos  y) .  6  z^. 

Again,  let  the  points  of  application  turn  around  the  axis  z,  parallel 
to  the  plane  xy\  the  components  parallel  to  the  axes  x  and  y  alone 
can  work,  since  the  patiis  will  be  perpendicular  to  the  components 
in  the  direction  of  z,  and  their  projections,  therefore,  zero.  The  ele- 
mentary work  in  this  case  will  be  tbund  in  the  fourth  term  of  Equa- 
tion (40),  and  equal  to 

[S  P  {x'  cos  /3  —  f/'  cos  a)]  (5  (p. 

Now  let  both  of  these  motions  take  place  simultaneousl}' ;  that  is,  let 
the  points  of  application  move  in  the  direction  of  the  axis  2,  and  also 
turn  about  that  line;  all  the  components  will  work,  because  the  paths 
will  be  oblique  to  their  directions,  and,  therefore,  have  projections  of 
measurable  values.  The  amount  of  elementary  work  of  the  extrane- 
ous forces  will,  in  this  case  be  found  in  the  third  and  fourth  terms 
of  Equation  (40),  and  equal  to 

[(S  P  cos  7)]  .5z^+[^P  {x'  cos  /3  -  y'  cos  a)]  .  5  9. 


MECHANICS     OF    isOLIDJ^. 


85 


The    same    reuun-ks    apply   to    motion    in    the    direction    of    and    about 
ca?h  of  the  other  axes. 

§  103. — The  rule  for  estimating  the  quantity  of  work  when  the 
motion  is  parallel  to  either  axis  or  to  a  i-ight  line  oblique  to  the 
three  axes,  is  simple ;  that  for  getting  the  work  during  motion  about 
an  axis,  is  not  so  obvious.  Let  the  motion  take  place  around  the 
axis  2;  and  considei',  first,  the  work  of  the  force  P.  The  two  compo- 
nents of  this  force,  viz.,  P  cos  /3  and  P  cos  «,  which  enter  the  fourth 
term  of  Equation  (40),  have  for  their  resultant  P  sin  y.  This  resultant, 
§81,  acts  in  a  plane  parallel  to  that  of  a;  y,  and,  therefore,  at  right 
angles  to  the  axis  z.  Denote  by  a^  the  angle  which  tliis  resultant 
makes  with  the  axis  x ;    then  will 


P  cos  a  z=  P  sin  y  .  cos  a^, 
P  cos  (3  =  P  sin  y  .  sin  a^ 

and  these  values  in  the  term  P  [x^  cos  /3  —  y'  cos  a),  give 

P  (x'  cos  (i  —  1/  cos  a)  =  P .  sin  y  (x'  sin  a^  —  y'  cos  aj 


(VO) 


(71) 


From  the  point  of  ap- 
plication in  of  P,  di'HW 
the  line  m  A'  perpendicu- 
lar to  the  axis  z ;  denote 
its  length  by  A',  and  its 
inclination  to  the  axis  .)• 
by  9'.  Multiply  and  ili- 
vide  Equation  (71)  by  // 
and  reduce  by  the  rel;i- 
tions 

x' 


Z 


/f 


-''., 

.      X 

-4^ 

/\^ 

/ 

I 

"^ 

V 

-,  =  COS  9' ;     —  =  sin  9 


then  will  result 

J°(a;'co3  0  —  y'  cos  a)  =  Psin  yh'  (sin  «, .  eos^'  — cosn, .  sin0'):=Psiiiy/('  sin  (a,  —  0'). 

Draw  from  A'  the  I'uo  A'  k'  perpendicular  to  the  direction  of  the  line 
P  m    (produced),  and  denote  its  length  by  k' ;  then  w"il] 

h'  sin  (a^  —  9')  =  A*', 


SG  ELEMENTS     OF     ANALYTICAL     MECHANICS. 

and  then  will  result 

F  (y  cos  (3  —  7/  cos  a)  =  P  sin  y  .  k',   ....     (72) 

and  the  same  for  the  forces  F',  F",  «&c.;    so  that  we  may  write,  omit 
ting  the  accent  from  k, 

E  F  {x'  cos  /3  ~  y'  cos  a)  =  2  P  .  sin  y  .  ^ ;    .     ,     .     (13) 

and  the  measure  of  the  elementary  work  due  to  rotation  about  the  axis 
z,  will  be  given  by  cither  member  of  the  Equation 

[E  F  {x'  cos  /3  -  3/'  cos  a)]  6 cp  =[E  F  sin  y.k]6(p.     .     (74) 

§  104. — So  that  in  estimating  the  work  due  to  rotation  alone  about 
the  axis  z,  each  force  is,  in  effect,  replaced  by  its  two  components,  the 
one  parallel,  the  other  perpendicular  to  that  line,  and  the  former  is 
neglected  because,  in  this  motion,  it  cannot  work. 

§  105. — The  quantity  of  work  obtained  by  multiplying  that  one  of 
the  two  components  of  a  force  which  is  perpendicular,  while  the  other 
is  parallel,  to  a  given  line,  into  the  perpendicular  distance  between  this 
line  and  that  of  the  force,  is  called  the  com2)onent  moment  of  the  force 
ill   reference  to  the  line. 

§  106. — The  line  in  reference  to  which  the  moment  is  taken,  is 
called,  in  genei'al,  a  component  axis;  the  perpendicular  distance  from 
the  axis  to  the  line  of  direction  of  the  force,  is  called  the  lever  arm  of 
the  force ;  and  the  extremity  of  the  lever  arm  on  the  axis  is  called  a 
centre  of  the  moment. 

When  the  direction  of  the  force  is  perpendicular  to  the  axis,  thr- 
latter  is  called  the  moment  axis  of  the  force.  In  this  case  the  compo- 
nent parallel  to  the  axis  becomes  zero,  and  the  normal  component  the 
force  itself. 

The  moment  of  the  resultant  of  several  conipoiicnt  forces  is  called 
the  resii/ldiit  ■moment.  The  moments  of  the  couipouent  forces  are  called 
component  moments. 

§  107. — Changing  da;)  into  d  cp  in  Equation  (74),  we  may  write 

[S  F  {x'  cos  |3  -  y'  cos  a)]  d(p=:[l.  F    sin  y  .  k\  d  (p   .     .     (74) 


MECHANICS    OF    SOLIDS.  87 

or 

yfs  P  {x'  cos  p-y'  cos  «)]  d:p  =J[l  P .  sin  y  .  k]  d  9    .     (74)' 

Whence  it  appears,  tliat  the  elementaiy  quantity  of  work  a  force  will 
perforin  during  the  motion  of  its  point  of  application  about  an  axis,  is 
equal  to  the  'product  of  the  moment  of  the  force  into  the  differential  of 
the  'path  described  at  the  uniCs  distance  from  the  axis.      t^  e  er  cra^    <p  uj    ou 

§  108. — The  whole  quantity  of  work  will  result  from  the  integration 
of  Equation  (74)'  between  limits.  In  this  integration  t\\o  cases  may 
arise,  viz. ;  either  the  moment  may  be  constant,  or  it  may  be  variable. 
In  the  first  case,  the  quantity  of  work  is  obtained  by  multiplying  the 
constant  moment  into  tlie  path  described  by  a  point  at  the  unit's  dis- 
tance from  the  axis.  In  the  second,  the  force  may  be  constant  and 
the  lever  arm  variable;  the  force  variable  and  the  lever  arm  constant; 
or  both  may  be  variable,  and  in  such  way  as  not  to  make  their  prod- 
uct constant.  In  all  such  cases,  relations  between  the  intensity  of  the 
force,  its  lever  arm,  and  the  path  desci'ibed  at  the  unit's  distance,  must 
be  known  in  order  to  reduce,  by  elimination,  the  second  member  of 
Equation  (74)'  to  a  function  of  a  single  variable. 

These  remarks  are  equally  true  of  the  forces  of  inertia.  The  intensi- 
ties of  these  depend  upon  the  masses  of  the  material  elements  and  their 
degree  of  acceleration  or  retardation;  their  points  of  application  are  on 
the  elements  themselves;  the  elementary  arc  described  at  the  unit's  dis- 
tance is  the  same  for  both  sets  of  moments,  and  its  value  depends  upon 
the  distribution  of  the  material   with  reference  to  the  axis  of  motion. 

The  moments  of  the  forces  which  urge  a  body  to  turn  in  opposite 
directions  about  any  assumed  axis  must  have  contrary  signs. 

The  sign  of  P  sin  y  k\  or  its  equal  P  cos  /3  .  x'  —  P  cos  a  .  y',  de- 
pends upon  the  angles  which  the  direction  of  the  foi'ce  makes  with  the 
axes,  and  upon  the  signs  and  relative  values  of  the  co-ordinates  of  the 
point  of  application. 

Let  the  angles  which  the  direction  of  any  force  makes  with  the 
co-ordinate  axes  be  estimated  from  the  positive  side  of  the  origin  ; 
then,  if  the  angles  which  this  direction  makes  with  both  axes  be 
acute,  and   the   point   of  ajiplication    lie   in   the  first  angle,  P  cos  ^  .x 


88  ELEMENTS    OF    ANALYTICAL    MECHANICS. 

and  P  COS  a  .  y',  will  be  positive,  and  if  tlie  first  of  these  product! 
exceed  the  second,  the  moment  will  he  positive ;  but  if  the  lattei 
be  the  greater,  the  moment  will  be  negative.  The  same  remarks 
apply  to  the  other  axes. 


COMPOSITION     AND     RESOLUTION     OF     MOMENTS. 

§  109. — The  forces  being  supposed  to  act  in  any  directions  whatever, 
join  the  point  of  applicatLon  of  the  resultant  R  and  the  origin  by 
a  right  line,  and  denote  its  length  by  //.  Multiply  and  divide  each 
of  the  Equations   (44)  by  //,   and  reduce  by   the  relations, 

~  =  cos  r,      ^--^7   -S^-^A- 

y  2 

;^=cos|, 
■^  =  cos  s, 

in    which   ^,    g   and    £,   denote    the    angles    which    the    line    H   makes 
with  the  axes  .r,   ij   and  £■,  respectively ;   then  will 

R.  H .  (cos  h  .  cos  ^  —  cos  a .  cos  |)  =  i^,    ~| 

R .  H.  (cos  a  .  cos  s  —  cos  c  .  cos  ^)  =  M,    \-    .     .     .    (75) 

R  .  H .  (cos  c  .  cos  f  —  cos  h  .  cos  s)  z=  iV.   j 

Squaring  each  of  these   Equations  and   adding,   we  find 

C     cos^  b  .  cos^  ^  —  2  cos  b .  cos  a  .  cos  ^ ,  cos  |  +  cos^  a  .  cos-  ^  "^ 
R^  .  JI~  -\  +  cos^  a  .  cos^  s  —  2  cos  a .  coS  c  .  cos  s  .  cos  ^  +  cos^  c  .  cos-  ^   r 

I  -f-COS^  C  .  COS^  g  —  2  cos  b  .  cos  O  .  cos  ^  .  cos  £   +  cos-  b  .  COS-  £  J 

=  Z2  +  i/J  +  iV^2 (7(5) 

But 


cos^  a  +  cos^  b  +  cos^  c  =  1, (77) 

cos2  ^   4-  cos^  g  +  cos2  £  =  1 , (78) 

COS  a  .  cos  ^  +  cos  i  .  cos  ^  +  cos  c  .  cos  s  —  cos  cp,  .     (79) 


MECHANICS     OF     SOLIDS.  89 

the  angle  (p,  being  that  made  by  the  Tine  //,  with  the  direction  of 
the  resultant. 

Collecting    the    co-efficients  of    cos^  «,    cos^  i,   cos^  c,   and   reducing 
by  the  following  relations,  deduced   from   Equation     (78)  ;   viz.  : 

cos^  s  +  cos2  1  =  1   —  cos^  ^, 
cos^  ^  +  COS"  s  =  1  —  cos-  I, 

COS^  f   +   COS^  ^   =:    1    —   COS^  £, 

we  find, 

i2^ .  JI~ .  [1  —  (cos  a .  COS  ^  +  COS  b  .  cos  ^  -f  cos  c  .  cos  sY']=L^-{-3P-\~a'^^  ; 

from  Equation   (70), 

1  —  (cos  a  .  cos  ^  +  cos  b  .  cos  |  +  cos  c ,  cos  s)^  =  1  —  eos^  cp  =  sin^  9  ; 

which  reduces  the  above  to 

E^.H^.  Sin2  9    rr:    Z2    +    J/2    +    jV\ 

But  i/2  _  gii;^3  (p  jg  ^};(g  square  of  the  perj^endicular  drawn  from  the 
origin  to  the  direction  of  the  resultant ;  it  is,  therefu-e,  the  square 
of  the  lever  arm  of  the  resultant  referred  to  the  origin  as  a  centre 
of  moments.  Denoting  this  lever  arm  by  IC,  .Ave  have,  after  taking 
the    square   root, 

R.K^  V"iF+  J/2  +  ^2 (80) 

That  is  to  say,  tfic  resultant  moment  of  any  system  of  forces  is  eqifol 
to  the  square  root  of  tl/e  sum  of  the  squares  of  tJie  sums  of  the  com- 
ponent moments,  taken  in  reference  to  any  three  rectangular  axes  through 
the  point  assumed  as  the  centre  of  moments. 

§  110.— Dividing    the    first  of  Equations    (75),  bV    Equation   (SO), 
we  find, 

IT  (cos  b  .  cos  ^  —  cos  a  .  cos  ^)  L 


K  ^L'  +  i/2  +  N2 

The  effect  of  a  force  is,  §77,  independent  of  the  position  of  its 
point  of  application,  provided  it  be  taken  on  the  line  of  direction. 
Let   the  point  of  application   of  7?,   be    taken   at  the  extremity    of  its 


90  ELEMENTS     OF    ANALYTICAL    MECHANICS. 

lever  arm,    then  will  II  coincide  with  and  be   equal  in  length  to  K 
Z,  and  I  will  become  the  angles  which  the  lever  arm    makes  with  th« 
axes  X  and  y,    respectively,    and    the    well    known    relation    obtained 
from    the    formulas   for   the    transformation    of  co-ordinates   from   one 
set  of  rectangular  axes  to  another,  will   give 

cos  Gj  =3  cos  h  .  cos  ^  —  cos  a  .  cos  | ; 

in  which  9^  is    the    angle    the    resultant    axis    makes   with    the    axis  z  ; 

whence,* 

L 
cos  0,  = _______ (81) 

VL'  +  M'  +  N' 

In  the  same  way,  denoting  by  6^  and  G,  the  angles  which  the 
moment  axis  of  R  makes  with  the  co-ordinate  axes  y  and  x  respec- 
tively, will 

cos  G,  =  =1 (82) 

N 

cos  0,  =  —3=^=::= (83) 

whence  we  conclude  *that,  the  cosine  of  the  angle  zvhich  the  resultant 
axis  makes  with  any  assumed  line  is  equal  to  the  sum ^of  the  moments 
of  the  forces  in  reference  to  this  line  taken  as  a  component  axis  divided 
by  the  resultant  moment. 

§  111. — Multiplying  Equation  (81)  by  Equation  (80),  there  will 
result, 

E.K.coiiQ,  =  L (81) 

which  shows  that  the  component  moment  of  any  system  of  forces  in 
reference  to  any  ohlique  axis  is  equal  to  the  2'^>'oduct  of  the  resultant 
moment  of  the  system  into  the  cosine  of  the  angle  betiveen  the  resultant 
and  com2)onent  axes. 

For  the  same  system  of  forces  and  the  same  centre  of  moments, 
it  is  obvious  that  R  and  K  will  be  constant ;  whence,  Equation  (80), 
the    sum    of    the    squares    of    the    sums    of  the    moments    in   reference 

*  See  Appendix,  No.  I. 


MECHANICS     OF     SOLIDS.  9] 

to  any  three  recianaular  axes  through  the  centre  of  moments  taken 
as  component  axes  is  a  constant  quantity/.  Also,  since  the  axis  g 
may  have  an  mfinite  inimber  of  positions  and  still  satisfy  the  con- 
dition of  making  equal  angles  with  the  resultant  axis  we  see 
Equation  (84),  that  the  sum  of  the  moments  of  tlie  forces  in  reference 
to  all  component  axes  tvhich  make  equal  angles  lolth  the  resultant 
axis  will  be   constant. 

§112.— Denote  by  d(,,J",  (>'",  the  angles  which  any  component 
axis  makes  with  the  co-ordinate  axes  z,  y  and  a;,  respectively,  and 
by  5  the  angle  which  the  component  and  resultant  axes  make  with 
each   other,  then  will 

cos  d  =  cos  9, .  cos  &,  +  cos  9^ .  cos  ^^  -f  cos  9^  .  cos  (\^  ; 

multiplying  both  members  by  R .  K,  we  have 

R . K.  cos 6  —  R. K.  cos  9, . cos  ^,  +  7? . K .  cos 9^, cos ^^■\-R,K. cos 9^ .  cos ^^ . 

But,  Eouatiou  f84), 

R  .  K .  cos  9j  =  Z, 
/^. /iT.  cos  9„=  if,      . 

/2.  A^.  cos  9^=  iY; 

which  substituted  above,  gives 

-R. /i^.  cosd  =:  i^.cos(),  +  i/.  cos^y  + -Y.  cos^^     .     .     (85) 

That  is  to  say,  tlie  component  moment  in  reference  to  any  assumed  com- 
2)onent  axis,  is  equal  to  the  sum  of  the  2)f'oducts  arising  fvm  multiplying 
the  sum  of  the  moments  in  reference  to  the  co-ordinate  axes,  by  the 
cosines  of  the  angles  which  the  direction  of  the  component  axis  makes 
with   these  co-ordinate  axes,  rcsp>ectlvcly. 

TRANSLATION    OF   EQUATIONS    {A)    AND    (5). 

§  Ho. — Equations  (J)  and  [B)  may  now  be  translated.  They  express 
the  conditions  of  equilibrium  of  a  system  of  forces  acting  in  various 
directions  and  upon  different  points  of  a  solid  l:ody.  These  f-ondi- 
tions   are    six    in   number ;  viz. : 


92  ELEMENTS     OE    ANALYTICAL    MECHANICS. 

1. —  The  algebraic  sum  of  the  couiponcnis  of  the  foxes  in  each  of 
any    three   rectangular   directions   must  be  separately   equal   to   zero ; 

2. — The  algebraic  sunt,  of  the  moments  of  the  forces  taken  in  refer- 
ence to  each  of  three  rectangular  axes  draivn  through  any  assumed 
centre  of  moments,  must  be   seixifutely    equal   to  zero. 

If  the  extraneous  forces  be  in  equilibrio,  the  terms  which  measure 
the  forces  of  inertia  will  disappear,  and  these  conditions  of  ecjuilibiium 
will    be   expressed   by 


2  P .  cos  a  =  0, 
2  P  cos  /3  =  0, 
2  P.  cos  7  =  0  ;_ 


{A)' 


2  P .  {x'  cos  (3  —  y'  cos  a)  =  0,  ^ 

2  P.  {z'.  cos  a  -  x'  cos  7)  =  0,  I      .     .     .     {B)' 

2  P.  [if  cos  y  —  z'  cos  /3)  =  0.  J 

The  above  conditions,  which  relate  to  the  most  general  action 
of  a  system  of  forces,  are  qualified  by  restrictions  imposed  upon 
the    state    of  the   body. 

1 114. — If  the  body  contain  a  fixed  -pointy  the  origin  of  the  mova- 
ble co-ordinates,  in  Equation  (40),  may  be  taken  at  this  point ;  in 
which  case  wo  shall  have, 

^x,  =  0,  . 

^V,  =  0, 
(J  .;  =  0  ; 

ami  it  will  only  be  necessary  that  the  forces  satisfy  Equations 
(/j),  these  being  the  co-efficients  of  the  indeterminate  qiiantities  that 
do  not  reduce  to  zero.  Hence,  in  the  case  of  a  fixed  point,  the 
sum  of  the  moments  of  the  forces^  taken  in  reference  to  each  of  three 
rectangular  axes,  2Jassing  through  the  point,  must  sep)arately  reduce  to 
zero. 

Should    the   system    contain    tico  fixed  points,  one  of  the  axes,   as 


MECHANICS     OF    SOLIDS.  93 

that  of  re,  may  be  assumed  to  coincide  with  the  line  joining  these 
points,  in   Nvhich   case,   there  will  result  in   Ecjuatioii  (40), 

o.v^  :=  0,  ^^  =  0, 
Sjj^  =  0,  S-^  =  0. 
Sz,  =  0, 

and  it  will  only  be  necessary  that  the  forces  satisfy  the  last  Equ.a- 
tion  in  group  (/>) ;  or  that  (he  sum  of  the  moments  of  the  forces  in 
reference  to  the  line  johiing  the  fixed  2^oints,   reduce  to  zero. 

If  the  system  be  free  to  slide  along  this  line,  ox^  will  not  reduce 
to  zero,  and  it  will  be  necessary  that  its  co-efficient,  in  Equation 
(40),  reduce  to  zero  ;  or  that  tlie  cdgehraic  sum  of  the  components  of 
the  given  forces  'parallel  to  the  line  joining  the  fixed  points,  also  reduce 
to  zero. 

If  three  points  of  the  system  be  constrained  to  remain  in  a 
fixed  plane,  one  of  the  co-ordinate  planes,  as  that  of  xy,  may  be 
assumed  parallel  to  this  plane ;    in  which  case, 

8z^  =  0, 
O'icr  =  0, 
S-\.  =  0; 

and  the  forces  must  satisfy  the  first  and  second  of  Equations  [A) 
and  the  first  of  {B)';  that  is,  the  algebraic  sum  of  the  conqwnents 
of  the  given  forces  jiarallel  to  each  of  two  rectangular  axes  parallel  to 
the  given  p)lane,  must  separately  reduce  to  zero,  and  the  sum  of  the 
moments  in  reference  to  an  axis  jyerjjendicular  to  this  j^lane  must  reduce 
to  zero. 

ce>:tre   of   GKAvmr. 

§115. — Gravity  is  the  name  given  to  that  force  which  urgc^  aii 
bodies  towards  the  centre  of  the  earth.  This  force  acts  upon  every 
particle  of  matter.  Every  body  may,  therefore,  be  regarded  as 
subjected  to  the  action  of  a  system  of  forces  whose  number  is  equal 
to  the  number  of  its  particles,  and  whose  points  of  application  have, 
with  respect  to  any  system  of  axes,  the  same  co-ordinates  as  these 
particles.    ( 


9-i  ELEMENTS     OF     ANALYTICAL    MECHANICS. 

The  weight  of  a  body  is  the  resultant  of  this  system,  or  the 
resultant  of  all  the  forces  of  gravity  which  act  iqyon  it.  and  is  equal, 
in  intensity,  but  directly  opposed  to  the  force  which  is  just  sufficient 
to  support  the   body. 

The  direction  of  the  force  of  gravity  is  perpendicular  to  the 
earth's  surface.  The  earth  is  an  oblate  spheroid,  of  small  eccentri- 
city, whose  mean  radius  is  nearly  four  thousand  miles;  hence,  as  the 
directions  of  the  force  of  gravity  convei'ge  towards  the  centre,  it  is 
obvious  that  these  directions,  when  they  appertain  to  particles  of 
the  same  body  of  ordinary  magnitude,  are  sensibly  parallel,  since 
the  linear  dimensions  of  such  bodies  may  be  neglected,  in  compari- 
son with  any  radius  of  curvature  of  the  earth. 

The  centre  of  such  a  system  of  forces  is  determined  by  Equa- 
tions (62),  §94,    which  are 

P'.^.'  +  p"a;"  4-  P'"x"'  4-  &c. 

5 


Z// 


P'   +  P"  +  P'"  +  &c. 

P'y'  +  P"y"  +  P"'y"'  +  &c. 
P'  +  P"  +  P'"  +  &c. 

P'z'  +  P"z"  +  P"'z"'  +  &c. 
P'  +  P"  +  P'"  +  &c. 


(86) 


in  which  x^  y^  z^,  are  the  co-ordinates  of  the  centre ;  P',  P".  6zc., 
the  forces  arising  from  the  action  of  the  force  of  gravity,  that  is, 
the  weights  of  the  elementary  masses  m'.  m",  &c..  of  which  the 
co-ordinates  are  respectively  x'  y'  z'.    x"  y"  z",  &c. 

This  centre  is  called  the  centre  of  gravity.  From  the  values  of 
its  co-ordinates,  Equations  (86),  it  is  apparent  that  the  position  of 
this  point  is  independent  of  the  direction  of  the  force  of  gravity  in 
reference  to  any  assumed  line  of  the  body;  and  the  centre  of  gravity 
of  ;t  l)ody  may  be  defined  to  be  t'lat  ^;(;i«<  through  which  its  zocighl 
always  passes  in  whatever  VHiy  the  body  may  be  turned  in  regard  to 
the  direction  of  the  force  of  gravity. 

The  values  of  /",  P",  &c.,  being  regarded  as  the  weights  iv\  xo", 
&c.,  of  the  elementary  masses  m',  m",  &c.,  we  have,  Equation  (1), 

P'  z=w'  =  m'g';    P"  =  w"  =  m"  g"  ;    P'"  =  to'"  =  m'"  g'"  ;   i-^'c, 


MECHANICS    OF    SOLIDS. 


V6 


and,   Equations  (86), 


m'  ff'x'  +  m"  r/"x"  +  m'" g 


in  ,,111  ^ni 


+    &C. 


vi' y'  +  m"  y"  +  m'" g'"  +  &c. 


2// 


m'//  +  m"y"  y"  +  m'"/ 


m'/  +  w"^"  +  m"'g"'  -f  &c. 


(87) 


§116. — It  will  be  shown  by  a  j^rocess  to  be  given  in  the  proper 
place,  that  the  intensity  of  the  force  of  gravity  varies  inversely  as 
the  square  of  the  distance  from  the  centre  of  the  earth.  The  distance 
from  the  surface  to  the  centre  of  the  earth  is  nearly  four  thousand 
miles  ;  a  change  of  half  a  mile  in  the  distance  at  the  surface  would 
therefore,  only  cause  a  change  of  one  four-thousandth  part  of  its 
entire  amount  in  the  force  of  gravity;  and  hence,  within  the  limits 
of  bodies  whose  centres  of  gravity  it  niay  be  desirable  in  practice  lo 
determine,  the  change  would  be  inappreciable.  Assuming,  then,  the 
foi'ce  of  gravity  at  the  same  place  as  constant.  Equations  (87), 
become 

m'  x'  +  m"  x"  +  m'"  x'"  +  &e.   >, 


y< 


m'  + 

in" 

+ 

1VJ" 

+  &c. 

m' 

y'  +  wi 

"y"  + 

m" 

y'"  +  &c. 

m'  + 

m" 

+ 

in'" 

+  &c. 

m' 

z'  +  m 

'  ,." 

4- 

on'" 

z'"  +  &c. 

m'  +  he"  +  m'"  -\-  &c. 


(88) 


from  which  it  appears,  that  when  the  action  of  the  force  of  gravity 
is  constant  thi'oughout  any  collection  of  particles,  the  position  of  the 
centre  of  gravity  is  independent  of  the  intensity  of  the  force. 

§  117. — Substituting  the  value  of  the  masses,  given  in  Equation  (1)', 
Inere  \;i\\  result. 


y. 


v'd'x'  +  v"d"  x"  +  v'"  d'"  x'"  +  &c.   >| 
v'  d'  +  v"  d"  +  v'"  d'"  +  &c^         ' 

v'  d'  y'  +  v"  d"  y"  +  v"'  d'"  y'"  +  &c. 
v'd'  +  v"  d"  +  v"'u''^"+&e;        ' 

v'd'z'  +  v"  d"  z"  +  V"  d"'z"'  +  &c. 
v'  d'  +  v"  d"  +  v'"  d'"  +  &c.        ' 


(8Q) 


OP. 


ELEMENTS     OF     ANALYTICAL    MECHANICS. 


and  if  the    elements  be  of  homogenous    density  throughout,  we   shall 
have, 

d'  =  d"  =  d'"  =z  &c.  ; 

aud  Equations   (89)   become, 

v'x'  +  v"x"  -f-  v"'x"'  +  &c.   ^ 


y< 


v'  +  v"  +  v'"  +  &c. 

v'rj'  +  v"y"  +  v'"  y'"  -^  &c. 
v'  +  v"  +  v'"  +  &c,         ' 

v'  +  «;"  +  v'"  +  &c.        ' 


(90) 


wiience  it  follows,  that  in  all  homogeneous  bodies,  the  position  of 
the  centre  of  gravity  is  independent  of  the  density,  provided  the 
intensity  of  gravity  is  the  same  throughout. 

§  118. — Employing  the  character  2,  in  its  usual  signification,  Equa- 
tions (90),  may  be  written, 

2  (v.r)    ^ 


2(.) 


y<  = 


V  y)    , 


2(.) 

-  -    2  (.)  '  j 

and  if  the  system  be  so  united  as  to  be  continuous. 


(91) 


Vi  = 


_  J,,.  y.dV    K 


V 


Jv"     "  ■ 


dV 


(<J2) 


V 


§119. — If  the   collection   be   divided   symmetrically    by   the   plane 
ry,  then  will 

2  (Z)  2)  =  0, 


MECHANICS     OF    SOLIDS. 


9T 


and,  therefore, 


=  0 


hence,  the  centre  of  gravity  will  lie  in  this  plane. 

If,  at  the  same  time,  the  collection  of  elements  be  symmetrically 
divided  by  the  plane  x  2,  we  shall  have, 

2  {yy)  =  0, 

the  collection  of  elements  will  be  symmetrically  disposed  about  the 
axis  .r,  and  the  centre  of  gravity  will  be  on  that  line. 

Although  it  is  always  true,  that  the  centre  of  gravity  will  lie  in 
a  plane  or  line  that  divides  a  homogeneous  collection  of  particles 
symmetrically ;  yet,  the  reverse,  it  is  obvious,  is  not  always  true, 
viz. :  that  the  collection  will  be  symmetrically  divided  by  a  plane  or 
line  that  may  contain  the  centre  of  gravity. 

Equations  (92)  are  employed  to  determine  the  centres  of  gravity 
of  all  geometrical  figures. 

IV 

THE   CENTKE   OF   GRAVITY    OF   LINES. 

§  120. — Let  s  represent  the  entire  length  of  an  arc  of  any  curve, 
whose  centre  of  gravity  is  to  be  found,  and  of  which  the  co-ordi- 
nates   of  the   extremities   are   as',  y\  z\  and  x" ,  y",  z". 

To  be  applicable  to  this  general  case  of  a  curve,  included  within 
the   given   limits.  Equations  (92)  become 


„   xdx. 

'J  X 


1  +  5^  +  lil 

d  .f-         d  x^ 


y,  = 


y  d  X 

s 

/; 

V- 

dy'' 

d  x~ 

dz^ 
■^  dx^ 

z  d  x 

.V 

/:: 

V- 

dy- 
dx' 

dz^ 

"^  (/  x^ 

Y*) 


(93) 


98  ELEMENTS     OF     ANALYTICAL    MECHANICS, 

in  wliich 


II"   ^  V 


d  ?/2         d  z^ 

1  H —  -\ 

d  x^         d  x^ 


(94) 


Example  1. — Find   the  2^oisition  of  the  centre   of  gravity   of  a   right 
line.     Let, 


z  =  a'x  +  13', 

be  the  equations  of  the 
line. 

Diffez-entiating,  substi- 
tuting in  Equations  (94) 
and  (93),  integrating  be- 
tween the  proper  limits, 
and  reducing,  there  will 
result, 

^  If  ^     a. 


z 


~X 


y, 


x' 

+ 

x" 

2 

a 

{x' 

+ 

:.") 

2 

a' 

.(..' 

+ 

X") 

+  /3, 


''  =  ^"2 +  ^'' 

which  are  the  co-ordinates  of  the  middle  point  of  the  line  ;  x'  y'  z' 
and  x"  y"  s",  being  those  of  its  extremities ;  whence  we  conclude 
that  the  centre  of  gravity  of  a  straight  line  is  at  its  middle  point. 

Example  2. — Find  the  centre  of  gravity  of  the  2'>erimeter  of  a  polygon. 
This  may  be  done,  according  to  Equations  (90),  by  taking  the  sum 
ol  the  products  which  result  from  multiplying  the  length  of  each  side 
by  the  co-ordinate  of  its  middle  point,  and  dividing  this  sum  by  the 
length  of  the  perimeter  of  the  polygon.  Or  by  construction,  as  fol- 
lows : 

The  weights  of  the  several  sides  of  the  polygon  constitute  a  system 
of  23iii'''illel  forces,  whose  points  of  application  are  the  centres  of 
gravity  of  the  sides.  The  sides  being  of  homogeneous  density,  their 
weights  are  proportional  to  their  lengths.     Hence,  to  find   the   centre 


MECHANICS     OF    SOLIDS. 


99 


of  gravity  of  the  entire  polygon,  join  the  middle  points  of  any  two 
of  the  sides  by  a  right  line,  and  divide  this  line  in  the  inverse  ratio 
of  the  lengths  of  the  adjacent  sides,  the  point  of  division  will,  §  97, 
be  the  centre  of  gravity  of  these  two  sides ;  next,  join  this  point 
with  the  middle  of  a  third  side  by  a  straight  line,  and  divide  this 
line  in  the  inverse  ratio  of  the  sum  of  first  two  sides,  and  this  third 
side,  the  point  of  division  will  be  the  centre  of  gravity  of  the  three 
sides.  Continue  this  process  till  all  the  sides  be  taken,  and  the  last 
point  of  division  will  be   the  centre  of  gravity  of  the   polygon. 

Find  the  ijosition  of  the  centre  of  gravity  of  a  plane  curve. 
Assume  the  plane  oi  xy  to   coincide  with  the  plane  of  the  curve, 
in  which  case, 

dz  I  -^'^ 


and  Equations  (93)  and  (94)  become, 


dy^ 
dx^ 


'W  y^ 


V 


oW.   [/-. 


T'%'~-4-  '-: 


y, 


/"--xA^ 


/::-nA^- 


(95) 


d    L 


(96) 


Example  3, — Find  the   centre  of  gravity  of  a  circular  arc. 
Take    the    origin    at    the    centre    of   curvature,    and   the   axis    of  y 
passing   through    the   middle    point  of  the   arc.     The   equation  of  the 
curve   is, 

/2   -—   /7.2   —  or!^ 


y^   =   fl"   —  X", 


whence.. 


dy^ 
dx 


which  substituted  in  Equations  (95), 


100         ELEMENTS     OF     ANALYTICAL    MECHANICS, 
will  give  on  reduction, 

X,  =  0, 

a  ix'  +  x")  . 

y.  = S ' 

and   denoting   the   chord  of  the   arc  by  c  =  a;'  +  x'\ 

X,  =  0, 
a  c 

whence  we  conclude  that  the  cenh-e  of  gravity  of  a  circular  arc  is 
on  a  line  draivn  through  the  centre  of  curvature  and  its  middle  2^oint, 
and  at  a.  distance  from  the  centre  equal  to  a  fourth  2^>'oportional  to 
the   arc,  radius  and  chord. 

Examj)le  4. — Find  the  centre  of  gravity  of  the  arc  of  a  cycloid. 

The   radius  of  the  generating  icircle  being   a,  the  differential  equa- 
tion of  the  curve  is, 


dx 


y  .dy 


y/'2ay 


yi 


(«) 


the    origin  being  at  A,  and 

A  B  being  the  axis  of  a;.  ji 

Transfer  the  origin  to  (7, 
and  denote  by  x',  y'  the  new 

co-ordinates,  the  former  being  estimated  in  the  direction   CD,  and  the 
latter  in  the  direction  DA.     Then  will 

?/  =  3a  —  x\ 

t . 


X  =  a*  —  y 


and  therefore. 


dx  _  d]}' 
dy        dx' 


y/'lax' 


(«)' 


MECHANICS     OF     SOLIDS. 


101 


this,  in    Equations   (9G)  and   (95),  gives,  omitting    the    accent    on    the 
variables, 


V^*^  •  ^  f- 


J  X 


X  d  X 


y, 


II"  y'''\l-x 


Integrating  the  first  two  equations  between  the  limits  indicated, 
and  substituting  the  value  of  s,  deduced  from  the  first,  in  the  second, 
we   have,  "?     -■    ' '-    . 

s   =2  V2a(V«"  -  ^/x'\ 

-  1    ^"""^  -  ^""'^ 
""''?-'  s/x"    --^x'    ' 

and  from   the  third  equation  we  have,  after    integrating  by  parts, 

«!//  =  3  V'2  a  {y  ^/  x  -  f^dy)  ;   r- 

substituting   the  value   of  dy,    obtained    from    Equation  (a)',    and   re- 
ducing,  there  will   result, 

sy,=2^/2  a  {y^x  -  f^2a  -  x.dx), 

and  taking  the  integral  between  the  indicated  limits, 

6-y,  =  2-/27^[y(-v/^  -  V^)  +  |(^«  -  ^")^  -  f(2a  -a;')2]; 
hence,  replacing  s   by  Its  value,  and  dividing, 


:r-~ 


y,  =  y  -\-  f 


(2«  -  x"Y  -  {2  a  -  x') 


y  X'    —  y  X' 

Supposing  the  arc  '.o  begin  at   C,  we  have, 

x'  =  0, 
and, 


_  1  ^'f 
3  •*"  » 


,  =  y+  ;^"-  r(2a  -  x")^   -  2«  V^27l 


102  ELEMENTS     OF    ANALYTICAL    MECHANICS. 

If  the  entire  semi-arc  from  C  to  A  he  taken,  these  values  become, 

a;,  =  f  «, 

y>  =  «  (*  -  f)- 

Taking  the  entire  arc  A  C  £,  the  curve  will  be  symmetrical  with  res 
pect  to  the  axis  of  x',  and  therefore, 

y.  =  0; 

lience,  the  centre  of  gravity  of  the  arc  of  the  cycloid,  generated  by  one 
entire  revolution  of  the  generating  circle,  is  on  the  line  tvhich  divides 
the  curve  symmetrically,  ccnd  at  a  distance  from  the  summit  of  the  curve 
equal  to  one-third    of  its  height. 


THE    CENTEE    OF    GKAVITY    OP    SURFACES. 

1 121. — Let  Z  =  0,  be  the  equation  of  any  surface;  L  being  a 
function  of  xyz\  then  will  dxdy,  be  the  projection  of  an  element 
of  this  surface,  whose  co-ordinates  are  x  y  z,  upon  the  plane  xy;  and 
if  (3"  denote  the  angle  which  a  plane  tangent  to  the  surface  at  the 
same  point  makes  with  the  plane  xy,  the  value  of  the  element  itself 
will  be 


dx .  dy 

cos  ^"  ' 

But  the  angle  which  a  plane 
makes  with  the  co-ordinate 
plane  x  y,  is  equal  to  the 
angle  which  the  normal  to 
the  plane  makes  with  the 
axis  0,  and,  therefore, 


cos  ^"  =    i 


dL 

dz 


^W^W^W)- 


(97) 


MECHANICS     OF    SOLIDS. 
and  hence,  in  Equations  (92),  omitting  the  double  sign, 
dVz=dX'cly-w,,     ,     .     , 
ani  those  Equations  become, 

Xj  = 


«/y    J  X 


w  .  X .d  X . dy 


y, 


J  11      J  X 


to  ,  y  dx  .  dy 


vf  y     J  X 


w  .  z .dx .dy 


in  which, 


\j'      „x' 


s  =   V  =    f  n  f  n  iv.dx.dy; 

J    V      J   X 


w  being  a  function  of  .r,  y,  s. 

If  the  surface  be  plane,  the 
plane  of  x  y  may  be  taken  in  the 
surface,  in  which  case, 

^y  =  1, 
z  =  0, 

and  Equations  (99),  and  (100),  be-  ^- 

Gome, 


J"  J" 


y, 


r  f  1 

J y"  J x"  dy  .xdx 
s 

j,/'J,"  dx.yd 


y(iy 


103 


(98) 


(99) 


(100) 


ir-X 


(101) 


=  fn  f,f  f^-^  •  dy, 

J  'I      J  X 


.      (102) 


in    which    the    integral    is    to    be    taken    first    with    respect    to   y,  and 


104: 


ELEMENTS  OF  ANALYTICAL  MECHANICS. 


between  the  limits  y"  =  P  m"  and  y'  =  P  m' ;   then  in  respect  to  a-, 
between  the  limits  x"  =  AP"^  and  x'  —  AP'.     Hence 


/ 


."{y"  -y').xd'. 


hL"{y"^- 


{y"^-y'^)dx 


(103) 


«=  f"n{y"  -y')dx. 

,1  X 


(104) 


y'  and  y",  denoting  running  co-ordinates,  which  may  be  either  roots 
of  the  same  equation,  resulting  from  the  same  value  of  x,  or  they 
may  belong  to  two  distinct  functions  of  x,  the  value  of  x  being  the 
same  in  each.      For  instance,  if 

F    {xy)  =  0, 

be  the  equation  of  the  curve  n'  m"  n"  m\  it  is  obvious  that  between 
the  limits  x"  =  A  P"  and  x'  =  A  P',  every  value  of  .r,  as  A  P, 
must  give  two  values  for  y,  viz.:  y"  =  P m"  and  y'  =  Pni'.      Or  if 


F{xy)  =  0, 
F'  [xy)  =  0, 

be  the  equations  of  two  distinct 
curves  in"  a"  and  m'  n\  referred 
to  the  same  origin  J,  then  will 
y"  and  y'  result  from  these 
functions  separately,  when  the 
same  value  is  given  to  x  in 
each. 


Example  1. — Required  the  portion  of  the   centre    of  gravity    of  the 
area  of  a  triangle. 


MECHAXiCS     OF    SOLIDS, 


105 


Let  ABC,  be  the  triangle. 
Assume  the  origin  of  co-ordi- 
nates at  one  of  the  angles  A, 
and  draw  the  axis  y  jDarallel  to 
the  opposite  side  B  C.  Denote 
the  distance  A  F  by  x',  and 
suppose, 

y"  =  ax, 

y'  =  ^x, 

to    be   the   equatioiJ^    of  the    sides    ^1  C  and    ^-1 B,    respectively,    then 
will 

y"  —  y'  =  {(t  —  ^)  -i', 


and, 


I     (a  —  b)  x^dx        ^ 


=  3"' 


y/  = 


/       [ci  —  0)  X  a  X 

if  xr 


whence  we  conclude,  that  the  centre  of  gravity  of  a  triangle  is  on  a 
line  drawn  from  any  one  of  the  angles  to  the  middle  of  the  opposite 
side,  and  at  a  distance  from  this  angle  equal  to  two-thirds  of  the  lint 
thus  drawn. 

Example  2. — Find  the  centre  of  gravity  of  the  area  of  any  polygon. 

From  any  one  of  the  angles 
as  A,  of  the  polygon,  draw  lines 
to  all  the  other  angles  except 
those  which  are  adjacent  on  either 
side;  the  polygon  will  thus  be 
divided  into  triangles.  Find  by 
the  rule  just  given,  the  centre  of 
gravity  of  each  of  the  triangles ; 


/ 


106 


ELEMENTS  OF  ANALYTICAL  MECHANICS, 


join  any  two  of  these  centres  by  a  right  line,  and  divide  this  line  in 
the  inverse  ratio  of  the  areas  of  the  triangles  to  which  these  centres 
belong  ;  the  point  of  division  will  be  the  centre  of  gravity  of  these 
two  triangles.  Join,  by  a  straight  line,  this  centre  with  the  centre  of 
gravity  of  a  third  triangle,  and  divide  this  line  in  the  inverse  ratio 
of  the  sum  of  the  areas  of  the  first  two  triangles  and  of  the  third,  this 
point  of  division  will  be  the  centre  of  gravity  of  the  three  triangles. 
Continue  this  process  till  all  the  triangles  be  embraced  by  it,  and  the 
last  j^oint  of  division  will  be  the  centre  of  gravity  of  the  polygon  ; 
the  reasons  for  the  rule  being  the  same  as  those  given  for  the  deter- 
mination of  the  centre  of  gravity  of  the  perimeter  of  a  polygon,  it 
being  only  necessary  to  substitute  the  areas  of  the  triangles  for  the 
lengths  of  the  sides. 

Exami^le  3. — Determine    the   position    of  the    centre  of  gravity  of  a 
circular  sector. 

The  centre  of  gravity  of  the  sec- 
tor will  be  on  the  radius  drawn  to 
the  middle  point  of  the  arc,  since  this 
radius  divides  the  sector  symmetri- 
cally. Conceive  the  sector  C  A  B,  to 
be  divided  into  an  indefinite  number 
of  elementary  sectors ;  each  one  of 
these  may  be  regarded  as  a  triangle 
whose  centre  of  gravity  is  at  a  dis- 
tance   from    the    centre    C,    equal    to 

two-thirds  of  the  radius.  If,  therefore,  from  this  centre  an  arc  be 
described  with  a  radius  equal  to  two-thirds  the  radius  of  the  sector, 
this  arc  will  be  the  locus  of  the  centres  of  gravity  of  all  the 
elementary  sectors;  and  for  reasons  already  explained,  the  centre  of 
gravity  of  the  entire  sector  will  be  the  same  as  that  of  the  portion 
of  this  arc  which  is  included  between  the  extreme  radii  of  the  sector. 
Hence,  calling  r  the  radius  of  the  sector,  a  and  c  its  arc  and  chord 
respectively,  and  x^  the  distance  of  the  centre  3f  gravity  from  the 
centre   (7,  we  have, 

^r .^c  2      r .c 


MECHANICS     OF     SOLIDS. 


1(^7 


The  centre  of  gravity  of  a  circular  sector  is  therefore  on  the  radius 
draivn  to  the  middle  point  of  tlie  arc  of  the  sector,  and  at  a  distance 
from  the  centre  of  curvature  equal  to  tiuo-thirds  of  a  fourth  propor- 
tional   to   the    arc,  chord  and   radius  of  the    sector. 

Example  4. — Find  the  centre  of  gravity  of  a  drcular  segment. 

Assume  the  origin  at  the  centre  C, 
and  take  the  axis  x  passing  through  the 
middle  point  of  thQ  arc,  the  centre  of 
gravity  in  question  will  be  on  this  axis, 
and,   therefore, 

y,  =  0. 

Let  A  B  HA  be  the  segment,  and 


the  equation  of  the  circle,  the  origin  being 
at    the  centre   C,  then  will 


^.-^ 


ff 


y"  = 

■\/  (fi 

-^•2, 

/  =  - 

l/a2 

~X\ 

and, 

Equations 

(103) 

and  (104), 

X,    =z 

^r 

r 

x .  dx 

s 

x' 

3 

s 

% 

.     P^'       . 

/ti- 

^       x'\ 

s    being   the    area  of   the    entire    segment.      Denoting   the   chord  A  B 
by  c,  we  have, 

whence, 

c3 


'  ~  12. s' 

and  we  conclude,  that  the  centre  of  gravity  of  a  circular  segment 
is  on  the  radius  draxvn  to  the  middle  of  the  arc,  and  at  a  distance 
from  the  centre  equal  to  the  cube  of  the  chord,  divided  by  twelve 
times    the   urea    of  the   segment. 


108 


ELEMENTS    OF    ANALYTICAL    MECHANICS. 


Replacing   the    v^alue   of  s,  and    supposing    x'    to   be    zero,  in  which 
case    the    segment  becomes   a   semicircle,  we   shall  find, 


4  a 


§  122. — If  the   surflice   be    one  of  revolution,  about   the    axis  x  for 
instance,  it  will  be    symmetrical  with  respect  to  this  axis  ;  hence, 


1/^  =  0 


0 


and  if  F{xy)  =  0,  be  the  equation  of  a  meridian  section  in  the 
plane  .ry,  then  will  the  area  of  an  elementary  zone  comprised  be- 
tween   tw'O   planes    perpendicular    to    the    axis  of  revolution   be. 


and  therefore,  Equations    (92), 


1  ,  ^^y"^   J 


Example  1. — Find 
the  jJosi7w;t  of  the 
centre  of  gravity  of 
a  right  conical  sur- 
face. 

The  equation  of 
tlic  element  in  the 
jthmo  X  I/,  is,  assum- 
ing the  origin  at  the 
vertex, 

hence, 


2  tt  I  f,  ax^  dx  -x/l  -\-  o^ 

J  X 

t,  = 

2  ■TT  /  ,,  ax  dx  -y/  \  +  a- 


=   X". 


(105) 
(100) 


MECHANICS     OF    SOLIDS. 


103 


Example  2. — Required  the  posi- 
tion of  the  centre  of  gravity  of 
a   spherical   zone. 

Assuming  the  origin  at  the 
centre,  the  eqiuation  of  the  me- 
ridian  curve   is, 

-*2    —   /7.2  . 


y. 


whence, 


ydy  - 

dy^ 
dx^ 


xdx, 


and. 


X  d.  X 


J  X 


ad  X 


X  "  —  .r  - 


2  {x"  -  x') 


x"  +  x' 


Hence,  the  centre  of  gravity  of  a  spherical  zone,  is  at  the  middle 
point  of  a  line  joining  the  centres  of  its  circular  bases.  And  in  the 
case  of  one    base    it  is   only   necessary  to   make   x"  =  a,  lohich  gives, 

x'   -{-  a 

So  that  the  centre  of  gravity  of  a  zone  of  one  base  is  at  the  middle 
of   the   ver-sine  of  its  meridiem  curve. 

THE    CENTKES     OF     GEAYITY    OF    VOLUMES. 

§123. — When  it  is  the  question  to  determine  the  centre  of  g!-avily 
of  the   volume  of  any  body,  we   have 

dV  =  d X  .  d y  .  d z, 
and  Equations  (92)  become. 


px'      pyf      pz' 

,t    /  ,r    If,  x.dy.dz.  dx 

J  X        J  V         J  z 


V 


no  ELEMENTS     OF    ANALYTICAL    MECHANICS. 

ynx'      py'      pz' 
u       ,,        ,,y-dy.dz.dx 
X       J  y       J  z 


y,  =-- 


V 


y->x'     py'     pz' 
n   l  ,t    I  „z.dy  .dz.d: 
X     J  y     J  z 


V 


and. 


yfx'     py'     pz' 
rr         ,r     f   n  dy  .dz.d'. 
X     J  y     J  z 


In  which  the  triple  integral  must  be  extended  to  include  the 
entire  space  embraced  by  the  surface  of  the  body ;  this  surface 
being   given   by  its  equation. 

If  the  volume  be  symmetrical  with  respect  to  any  line,  this  line 
may  be  assumed  as  one  of  the  co-ordinate  axes,  as  that  of  x ;  in 
which  case,  if  X  represent  the  area  of  a  section  perpendicular  to  this 
axis,  and  x,  its  distance  from  the  plane  yz,  then  will  Xdx^  be  an 
elementary  volume  symmetrically  disposed  in  regard  to  the  axis  x, 
and  Equations  (92),  become 

px' 

I  ,,  Xxd X 
^.---^ » (lOT) 

S'.  =  0, 
z,  =  0, 
and, 

F=  Jl"  Xdx (108) 

Exami^le  1. — Find  the  position  of  the  centre  of  gravity  of  a  semi- 
ellipsoid,  the  equation  of  w^hose  surface  is 

,j.2  0/2  £1 

The    semi-axes  of  the  elliptical  section  parallel  to  the  j)]ane  y  z,  are. 


y  =  B\/  1  -^2' 


MECHA.NICS     OF    SOLIDS.  Ill 

whence, 

and,  Equations  (107)  and   (108), 

f\BC  (\  -^^  xdx 
X,  =  =.  --  A. 


If  the  figure  be  one  of  revolution  about  the  axis  of  a:,  then,  denoting 
by 

F{xy)  =  0, (loy) 

the  equation  of  the  meridian  section  by  the  plane  x  y,  will 

X=i(y\ 

and  Equations  (107)  and  (108),  may  be  written, 


/  „    It y"^ X  dx 

J  X 


•'^/  = V ' (11^) 

V=    ri'^y'^dx (Ill) 

J  X 

Example  1. — Required   the  i^osition    of  the   centre   of   gravity    of  a 
'paraboloid  of  revolution. 

In  this  case,  Equation  (109), 

I'[xy)  =  y^  —  2px  =  0, 
whence, 

F  =  2  ifp  f    xd  X, 

V  a 

^nf  2i  I   x^  d  X        2 

X,  = =  —  a. 

2  *  p  f   X  d  X 

J  a 


112  ELEMENTS     OF     AJs^ALYTICAL    MECHANICS. 

ExariijAt  2. — Required    the  2^osi(ion  of  the   centre   of  gravity   of  the 
volume  of  a  spherical  segment. 


whence, 


F{xy)  =2/'  +  ^'-«'  =  0, 

V  =  'g  r'ia^  —  x^)dx 

"'  /  ,,  i!^"  —  X")  .X  .dx 


X.   — 


*  /  ^  (a2  —  x-")  d  X 

or, 

3     \.t"-{:l(fi  -  x"'^)'X"^{\La^  —  ;r^2)n 

and  for  a  segment  of  one   base,   re"  =  f,  ^ 

^'  ~  T  '  2ri3  -  x'    (8a2  -  a;'2)' 

If  the  volume  have  a  plane  face,  and  be  of  such  figure  that  the 
areas  of  all  sections  parallel  to  this  face,  are  connected  by  any  law 
of  their  distances  from  it,  the  position  of  the  centre  of  gravity,  may 
also  be  found  by  the  method  of  single  integrals. 

Example  1. — Find  the  centre  of  gravity  of  any  pyramid. 

Find  by  the  method  explained,  the  centre  of  gravity  of  the  base 
of  the  pyramid,  and  join  this  point  with  the  vertex  by  a  straight  line. 
All  sections  parallel  to  the  base  are  similar  to  it,  and  Avill  be  pierced 
bv  this  line  in  homologous  points  and  therefore  in  their  centres  of 
gravity.  Each  section  being  supposed  indefinitely  thin,  and  its  weight 
acting  at  its  centre  of  gravity,  the  centre  of  gravity  of  the  entire 
pyramid  will,  §97,  be  found  somewhere  on  the  same  line. 

Take  the  origin  at  the  vertex,  draw  the  axis  x  perpendicular  to 
the   plane   of    the    base,    and    the    plane   xy    through   its    centre   of 


MECHANICS     OF     SOLIDS. 


113 


gi-avity ;  and   let  X  represent  any  section   parallel  to    the   base,  then 
will  Equations  (92)  become, 

px' 

I  ,,  Xxdx 


X.   = 


V 


y,  = 


ynx' 
nXy 

z,  =  0, 


dx 


and, 


fix' 

V  =  j^„  Xdx. 
Represent  by  A  the   base  of  the  pyratoid,  c  its  altitude,  and  let 

y  =z  ax, 

be   the  equation  of  the  line  joining  the  vertex  and  centre  of  gravity 
of  the  base. 


Then, 


and  for   any  frustum, 


A:X::c^:x\ 


'"'  Ax'^dx 


^  ~  h'    c' 

—rlffX^dx  _,        ^    ,,,  ,^ 

_   cV«  _    3    /x"^  —  x'^\ 


y, 


c 
a  A 


JJ' 


x'^dx 
x^  dx 


_    3       ^x"^  —  x'^\ 


— —  /   X" dx 

c^  Jxll 

and  for   the   entire  pyramid,  ftiake  x"  =  c,  and  x'  =  0,  which  give 


tC ,   —    .  c. 


y,  =  fac; 


114:         ELEMENTS     OF     ANALYTICAL    MECHANICS. 

whence  we  conclude  that  tJte  cevdre  of  gravity  of  a  pyramid  is  on 
the  line  drawn  from  the  vertex  to  the  centre  of  gravity  of  the  base, 
and  at  a  distance  from  the  vertex  equal  to  threcfourths  of  the  length 
of  this    line. 

The  same  rule  obviously  applies  to  a  cone,  since  tlie  result  is 
independent  of  the  figure  of  the    base. 

The  weight  of  a  body  always  acting  at  its  centre  of  gravity,  and 
in  a  vertical  direction,  it  follows,  that  if  the  body  be  freely  sus- 
pended in  succession  from  any  two  of  its  points  by  a  perfectly 
flexible  thread,  and  the  directions  of  this  thread,  when  the  body  is 
in  equilibrio,  be  produced,  they  will  intersect  at  the  centre  of  gravity, 
and  hence  it  will  only  be  necessary,  in  any  particular  case,  to  deter- 
mine this  point  of  intersection,  to  find,  experimentally,  the  centre 
of  gravity  of  a   body. 

THE    CENTKOBAKYC    METnOD. 

§124. — Kesuming   the    second  of  Equations  (95)  and  (103),  which 
are, 


in  which 


and 


in  which 


r'     ,         L        dy^ 

^Jj,ydxs^l+j^ 

y, ■ 


y,  = 


"  {'/'  —  y')  ^  ^ ; 

clearing    the    fractions    and    niultip-lylng    both    members    by     2  77,    we 

shall   have, 

• 

2it.y^s  =  £„2'^y    ^/d^"^  +  dy\      •     •     •     (112) 

2<y,s-=  £','.{y""~-y'^)dx     ....     (113) 


MECHANICS     OF     SOLIDS. 


115 


T}ic  second  uiember  of  Equation  (112)  is  the  area  of  a  surface 
geiieratecl  by  the  revolution  of  a  plane  curve,  whose  extremities 
are  given  by  the  ordinates  answering  to  the  abscisses  x'  and  x", 
about  the  axis  x.  In  the  first  member,  s  is  the  entire  length  of 
this  arc,  and  S'^^'y^  is  the  circumference  generated  by  its  centre  of 
gravity.  Hence,  we  have  this  simple  rule  for  finding  the  area  of  a 
figure   of    revolution,  viz.  : 

Multiphj  the  length  of  the  generating  curve  by  the  circumference 
described  by  its  centre  of  gravity  about  t/ie  axis  of  "-otation ;  the 
'product  will   be    the    required   surface. 

The  second  member  of  Equation  (113)  is  the  volume  generated 
by  a  plane  area,  bounded  by  two  branches  of  the  same  curve  or 
by  two  different  curves,  and  the  ordinates  answering  to  the  abscisses 
x'  and  a;",  about  the  axis  x.  s,  in  the  first  member,  is  the  generating 
area,  and  2'7ry^  the  circumference  described  by  its  centre  of  gravity. 
Hence,  this  rule  for  finding  the  volume  of  any   figure  of  revolution,  viz. : 

Multiply  the  generating  area  by  the  circumference  described  by  its 
centre  of  gravity  about  the  axis  of  rotation ;  the  jyroduct  loill  be  the 
volume   sought.  ^ 

Exam'ple   \.— Required  the  measure  of  the  surface  of  a  right  cone. 

Let  the  cone  be  generated  by  the 
rotation  of  the  line  A  B  about  the 
line  A  C.  The  centre  of  gravity  of 
the  generatrix  is  at  its  middle  point 
G,  and  therefore,  the  radius  of  the 
circle  described  by  it  wall  be  one- 
half  of  the  radius  C  B,  of  the  circu- 
lar base  of  the  cone.      Hence, 

BC.AB 


B 


^Ziiy^.s 


^'^  BC.AB. 


Example  2. — Find  the  volume  of  the  cone. 

The  area  of  the  generatrix   A  B  C\  h  }>  B  C .  A  C ;    and  the  radius 
of  the  circle  described  by  its  centre  of  gravity  is  -}  B  C.     Hejice, 


y/« 


I'TfBC 


BC  .AG 


BGKA  G 


116 


ELEMENTS     OF     ANALYTICAL    MECHANICS 


CENTRE     OF    EvERTIA. 

g  125. — When  the  elementary  masses  of  a  body  exert  their  forces 
of  inertia  simultaneously  and  in  parallel  directions,  they  must  expe- 
rience equal  accelerations  or  retardations  in  the  same  time,  and  the 
factor 

in  the  measures  of  these  forces,  as  given  in  Equation  (13),  must  be 
the  same  for  all.  Substituting  these  measures  for  P',  P",  &c.,  in 
Equations  (62),  we  fmd. 


'dfi 


dh 
'W 

dh 
'dF 


m  y 


2m 


■  2  m  z' 


2  m 


2  7/1  if 

2  m 


1.  mz 

2  m 


(114) 


Whence,  Equations  (88),  the  centre  of  inertia  coincides  with  the 
centre  of  gravity  when  the  force  of  gravity  is  constant,  both  being  at  the 
centre  of  mass.  In  strictness,  however,  the  centre  of  gravity  is 
always  below  the  centre  of  inertia;  for  when  the  variation  in  the 
force  of  gravity,  arising  from  change  of  distance,  is  taken  into 
account,  the  lower  of  two  equal  masses  will  be  found  the  heavier. 
And  in  bodies  whose  linear  dimensions  bear  some  aj)preciable  propor- 
tion to  their  distances  from  the  centre  of  attraction,  the  distance 
between  these  centres  becomes  sensible,  and  gives  rise  to  some  curious 
l^henomcna. 


M  P:  C  i !  A  X  I  ( '  S     OF     SOLIDS. 


117 


MOTION    OF    THE    CENTRE    OF    INEKTIA. 


§  126. — Substitute  in  Equations  (^1),  the  values  of  d'^  x,  (P  ij.  and  fp2, 
given  by  Equations  (34),  and  we  have,  because  dt  is  constant,  and 
d-x^^  d'^y^  and  d'^z^^  will  each  be  a  common  factor  for  all  the  elemen- 
tary masses, 


2  P  cos  a  —  i/ 


J-.r, 


1 


2  P  cos  /3  —  M 
2  P  cos  7  —  M- 


dfi  dt" 


dt^ 
d-^z. 


dl-^ 
1 


dfi  dfi 


2  m .  (^2  x'  =  0, 
•  2  wt .  fZ2  y'  =  0, 
2  m  .d"z'  =  0. 


in  which  M,  denotes  the  entire  mass  of  the  body,  being  equal  to  2  m. 
Denote  by  x,  y,  z,  the  co-ordinates  of  the  centre  of  inertia  referred 
to  the  movable  origin,  then,  Equations  (114), 

M.  X  —  ^mx', 
M .  y  z=  1  my\ 
M.  z  —  l.m  z\ 


and  differentiating  twice. 


M.  d^x  —  2  m  .  d\r',  ^ 
M.  dhj  =  2  ?«  .  c/2^', 
M .  dH  —  2  m  .  d-z', 


which  substituted  in  the   preceding  Equations,  give. 


2P.cos«-.l/.-^_i/_  =  0, 

2P.cos,g-J/.^    -M-%=0, 
dt^  dfl 


(A-, 


r/2. 


2P.cos,-^.-^-3/.-^.0, 


(11 5) 


(116) 


118 


ELEMENTS     OF     ANALYTICAL    MECHANICS. 


and  if  tlie    movable    origin    be    taken    at    the    centre    of  inertia,    then 
will, 

dH  =  0,     dfy  =  0,     (^2-  _  0  ; 

and   a-^ ,  y^ ,  s^ ,    will   become  the  co-ordinates  of  the  centre  of  inertia 
referred  to  the  fixed  origin,  and  we  have, 


2  P  ,  cos  a 
2  P.  cos /3 
2  P.  cos  7 


(117) 


Equations  which  are  wholly  independent  of  the  relative  positions 
of  the  elementary  masses  m',  m"  &c.,  since  their  co-ordinates  x',  y\ 
z' ,  &c.,  do  not  enter.  It  will  also  be  observed  that  the  resistance  of 
inertia  is  the  same  as  that  of  an  equal  mass  concentrated  at  the 
body's  centre  of  inertia. 

Whence  we  conclude,  that  when  a  body  is  subjected  to  the  action 
of  any  system  of  extraneous  forces,  the  motion  of  its  centre  of  inertia 
will  be  the  same  as  though  the  entire  mass  were  coneenti'ated  into 
(hat  point,  and  the  forces  applied  without  change  of  intensity  and 
direction,  directly  to  it. 

This  is  an  important  fact,  and  shows  that  in  discussing  the  motion 
of  translation  of  bodies,  we-  may  confine  our  attention  to  the  motion 
of  their  centres  of  in'ertia  regarded  as  material  points. 


EOTATION     AROUIsT)    THE    CENTRE    OF    INERTIA. 

§  127. — Now,  retaining  the  movable  origin  at  the  centre  of  inertia, 
substitr.tc  in  Equations  (//),  the  values  of  (/^.c,  rf^y,  and  <P-z.,  as  given 
by  Equations  (34),  and  reduce  by  the  relations, 

M  .x^:^'^  m  .  x'  =  0, 
M  .~y  =  ^iii.y'  —  0, 
ill .  £  =  2  m  .  z'  =  0; 


MECHANICS    OF    SOLIDS.  119 

and  we  have, 

—-■'  x' —  •  y'J  =  0, 

2  P.  (cos  a  .  s'  —  COS  7  .  x')  —  2  ??i  •  [-—  ■  z'  —  — ^  •  x'  J  =0,  [^  (1 18) 
SP.{cosy.y'  -  COS/3.2')  -2m.  [^  -  v'  --^•^')  =0; 


from  which  all  traces  of  the  position  of  the  centre  of  inertia  have 
disappeared,  and  from  which  we  infer  that  when  a  free  body  is  acted 
upon  by  any  system  of  forces,  the  body  will  rotate  about  its  centre 
of  inertia  exactly  the  same  whether  that  centre  be  at  rest  or  in 
motion. 

§128. — And  we  are  to  conclude,  Equations  (IIT)  and  (118),  that 
when  a  body  is  sul)jected  to  the  action  of  one  or  more  forces,  it  will 
in  general,  take  up  two  motions — one  of  translation,  and  one  of  rota- 
tion,  each  being  perfectly  independent  of  the  other. 

§  129.— Multiply  the  first  of  Equations  (117),  by  y^ ,  the  second  by 
x^ ,  and  subtract  the  first  product  from  the  second ;  also,  the  first  by 
z^  ,  the  third  by  x^ ,  and  subtract  the  second  of  these  products  from 
the  first ;  also  the  third  by  y^  ,  and  the  second  by  z^ ,  and  subtract 
the  second  of  these  products  from  the  first,  and  v>'e  hare, 

2(Pcos,/3)..r,-2(Pcosa).y-J/.  (^^ . .^.    _  ^ .  y^)  ^0,  " 

2  {P  cos  c.).  z-^{F  cosy).  x-M-  (-^ .  ..^  _  1^ .  .^  J  ^0,  [(119) 

2(Pcos7).y-2(Pcos;5)..-J/.  (^.y^_f|.,^)  ^0 


Equations  from  which  may  be  found    the  ■  circumstances  of  motion 
of  the  centre  of  inertia  about  the  fixed  origin. 


120  ELEMENTS     OF     ANALYTICAL    MECHANICS. 


MOTION    OF     TEANSLATION. 


§  130. — Eegarding  the  forces  as  applied  directly  to  the  centre  of 
inertia,  replace  in  Equations  (117),  the  values  DP.  cos  a,  DP.cosjS, 
and  2  P.  cosy,   hj  X,   Y,  and  Z,  respectively,  and  we  may  write. 


dfi  ' 


(120) 


from  which  the  accents  are  omitted,  and  in   which   x,  y,  and  z,   must 
be  understood  as  appertaining  to  the  centre  of  inertia. 


GEJSTEEAIi    THEOREM    OF    WORK,    \rELOCITY    AND    LIVING    FORCE. 

§131. — Multiply  the  first  of  Equations  (120)  by  2dx^  the  second 
by  2dy,  the  third   by  2dz,  add   and    integrate,  we  have 

2f{Xdx  +  Ydy  +  Zdz)  -  M.  ^^■^'"  +  df  +  dz"^    j^  c  =  0. 


But, 


whence. 


d^  ~~  Ifi 


dfi 


=     F2 


2f{Xdx  +  Ydy  +  Zdz)  -  M.V^  +  C  =  0     .     .     (121) 

The  first  term  is,  §101,  twice  the  quantity  of  work  of  the  ex- 
traneous forces,  the  second  is  twice  the  quantity  of  work  of  the 
inertia,  measured  by  the  living  force,  and  the  third  is  the  constant 
of  integration. 

If  the  forces  X,  Y,  Z,  be  variable,  they  must  be  expressed,  in 
functions    of    x,    y,    z,    before     the     integration     can    be    perfoimed 


MECHANICS     OF     SOLIDS.  121 

Supposing  this  latter  condition  fulfilled,  and  that  the  forms  of  the 
functions  are  such  as  make  the  integration  possible,  we  may  write, 

F{xyz)-^M.V^+C'=0,      ....     (122) 

and  between  the  limits  Xj  y,  z^    and   a;/  y/  2/  , 

F  C^/  V!  0  -  F  (-^V  y.  ^)  ^iM{V  -"  -  T7)   .     .  (123) 

whence  we  conclude,  that  the  quantity  of  work  expended  by  the 
extraneous  forces  impressed  upon  a  body  during  its  passage  from  one 
position  to  another,  is  equal  to  half  the  difference  of  the  living  forces 
of  the  body  at  these  two  positions. 

"VVe  also  see,  from  Equation  (123),  that  whenever  the  body 
returns  to  any  position  it  may  have  occupied  before,  its  velocity  will 
be  the  same  as  it  was  previously  at  that  place.  Also,  th;it  the 
velocity,   at  any  point,   is  wholly   independent  of  the  path  described. 

§  132.— If 

Xdx  +  Ydy  +  Zdz  =  0, 
the  extraneous  forces  will,   §101,   be  in   equilibrio,  and 


V 


/2.C" 


that    is,    the    velocity    will   be    constant,    and    the    motion,    therefore, 
uniform. 

§  133.— Again,  multiply  the  first  of  Equations  (118)  by  d  cp,  the  sec- 
ond by  d  ip,  the  third  by  d  zj  ;  add  and  reduce  by  the  rehitions  given 
in  Equations  (38) ;    we  find 

r,  ,    ,  ,       T,        ^  ,    ,  ,       T-,  ,  .  /iPx'.dx'   ,   d'y'dif  ,   d'^z  .dz\ 

xPcosadx  -f  sPcos/3rf?/4-S  Pcosydz  =ziii  {  r-5 1 r^  H rz —  I; 

\      dt  at  dt      / 

integrating   and   replacing   the   first   member   by  its   equal   in   Equation 
(68),  Ave  have 

Denoting  the  lever  arm  of  it  by  A',  the  vdooitN'  of  the  molecule  ;/;   in 
reference  to  the  centre  of  inertia  bv  /',  ttc,  and  tlic  arc  described   h\  a 


l22  ELEMENTS     OF     ANALYTICAL    MECHANICS. 

point  in  tlio  plane  of  tlie  i-esultant  R  and  of  its  lever  arm,  at  the  unn'a 
distance  from  the  centre  of  inertia,  by  s^,  we  have 


J Bdr^J R K. d s^ ; ^^■^,.^   ^  =  v',  &c. ; 


d.r''  +  dy''  +  dz'' 

whence 

fR.K.ds,  =  \^mv'+  C 

Adding  this  to  Equation  (121),  there  will  result 

2j\xd X  +  Yd y  +  Zd z)  +  'iJr . K. ds,^  M  V'+  E  m  y^+  C   (121)' 

From  which  it  appears,  that  tlie  quantity  of  woi'k  a  system  of  forces, 
applied  to  a  free  body,  Avill  perform,  in  giving  to  it  a  velocity  F,  will 
vary  with  the  distance  K  of  the  line  of  direction  of  the  resultant  from 
the  centre  of  inertia,  and  that  the  living  force  will  be  that  due  to  a 
motion  of  translation  of  the  centre  of  iii:;';,  increased  by  that  duo 
to  rotation  about  that  centre. 
j^  If  Equation  (121)  be  applied  to  each  oue  of  a  collection  of  elements 
»i,  'iii\  etc.,  there  will  be  as  many  equations  as  elements ;  and  denoting 
the  velocities  of  the  latter  by  v^  v\  itc,  we  have,  by  addition, 

2  ^f{Xd X  +  Yd y  +  Zd z)  =  1'  m  v'  -  C      .     .     (121)" 

Let  the  extraneous  forces  be  only  those  ai'ising  from  the  mutual  actions 
and  reactions  of  the  elements  upon  one  another.  If  the  elements  m 
and~77»^-h£_separated  bj'  the  distance  r,  and  their  co-ordinates  be  xyz 
and  x'y'z',  respcctrveTyT^tlienTAvill 

X  —  x                              y  —  _?/'                            z  —  z' 
•  cos  p  = ;  cos  y  = ; 


r 


,  X  —  X     ^  7/  —  ?/  Z  —  Z 

cos  a  = ;     cos  p  =  —  ' ^  ;     cos  y  = : 

r  r  r 

rtud  for  the  element  vi  we  have 

Xdx  +  Ydy  +  Zdz  =  F  {     ~  ^  dx  +  ^^^^  dy  +  ''-^^^dz\  ; 

\     /•  r  r  J 

for  the  clement  m^, 

X'dx'+Y'd7/  +  Z'dz'=-p{'^—^dx'-{-'-^^^dy'-\-^-^^dz'\- 

and  by  addition, 

Xc7  x+Ydyi-zdz+X(7  x  +  rd  :/'+z'ri  z'=-[{x-x-)d  {x-y)+{'j-v)  'n;/-v')+(.^-^')<^  i^-^ll 

T 


MECHANICS     OF    SOLIDS.  123 

But 

and  difterentiatiiici;, 

r  d  r  ={x-  x')  d  {x  -  X')  +  {>j  -  y')  d  (y  -  //')  +  (2  -  z')  d  {z  -  z') ; 
so    that    the    second    meniber    above    reduces    to  P dr\    and    Equation 

\/LfPdr  =  y.mv-—C (121)'" 


(121)"  to 


If  the  elements  be   invariably  connected   during  the  motion,  the  differ- 
entials of  r  will  be  zero,  and 

2  ni  v^  =  a 

This  is  called  the  conscrvuilon  of  livlnc/  force. 

STABLE    AND    UNSTABLE    EQUILIBRrUM. 

§  134. — Resuming  Equation  (123),  omitting  the  subscript  accents, 
and  bearing  in  mind  that  the  co-ordinates  refer  to  the  centre  of 
inertia,  into  which  Ave  may  suppose  for  simplification  the  body  to  be 
concentrated,  we   may  write, 

|.1/F'2  -  1J/F2  =  F{x'y'z')  -  F{xyz), 

in    v.hich 

F{xyz)  =  f{Xd.r  +   Ydy  +  Zdz), 
and 

dF{xyz)  =  Xdx  +  Ydy  +  Zdz. 

Now,  if  the  limits  x'  y'  z'  and  xyz  be  taken  very  near  to  each 
other,  then    will 

x'  zz:.  X  -\-  d  x  \     y'  —  y  -\-  dy;     s'  =z  z  -\-  dz; 
which   substituted    above,  give 

|l/7'2  _  IMV^  =  Fix  +  dx,  y  +  dy,  z  +  d£)  -  F{xyz), 

and   developing   by  Taylor's  theorem, 

(        Adx  +  Bd7/  4-  Cdz 
^  ^  (  +  A'  dx"-  +  B'  dy"-  +  &c.  +  D, 

in    which   D   denotes    the    sum    of    the    terms    involving    the   higher 
powers    of  dx^  dy  and  d  z. 


12-i  ELEMENTS     OF    ANALYTICAL    MECHANICS. 

If   I  i/ F  2  be    a    maximum  or    minimum,  then    will 

Adx  -^  Bdy  +  Cdz  zr^Q; (123)'- 

and   since 

Adx  +  Bdy  +  Cdz  =  dF{x7jz)  =  Xdx  +   Ydy  +  Zdz, 

we   have, 

Xdx  +   Ydy  +  Zdz  =  0. 

But  when  this  condition  is  fullilled,  the  forces  will,  Equation  (00), 
be  in  equilibrio  ;  and  we  therefore  conclude  that  whenever  a  body 
whose  centre  of  inertia  is  acted  upon  by  forceo  not  in  equilibrio, 
reaches  a  position  in  whieh  the  living  force  or  the  quantity  of 
work    is   a  maximum   or  minimum,  these  forces  will   be    in  equilibrio. 

And,  reciprocally,  it  may  be  said,  in  yeiicral,  that  when  the  forces 
are  in  equilibrio,  the  body  has  a  position  such  that  the  quantity  of 
action  will  be  a  maximum  or  minimum,  though  this  is  not  always 
true,  since  the  function  is  not  necessarily  either  a  maximum  or  a 
minimum  when  its  first  differential  co-eillcient  is  zero. 

§  135. — Equation   (123)' ,  being  satisfied,  we  have 

iMV'^  -  ^MV^  =  ±  {A'dx^-  +  B'dy'-  +  &c.  +  !))■•'  (124) 

The  upper  sign  answers  to  the  case  of  a  minimum,  and  the  lower 
to  a  maximum. 

Now,  if  V  be  very  small,  and  at  the  same  time  a  maximum,  V 
must  also  be  very  small  and  less  than  F,  in  order  that  the  second 
member  may  be  negative;  whence  it  appears  that  whenever  the  systeir 
arrives  at  a  position  in  which  the  living  force  or  quantity  of  work  i> 
a  maximum  and  the  .  system  in  a  state  bordering  on  rest,  it  cannot 
depart  far  fro7n  this  positipn  if  subjected  alone  to  the  forces  which 
biouf;lit  il  tnere.  This  position,  which  we  have  seen  is  one  of  cqui- 
lii)iium,  is  called  a  position  of  stable  cqinUbrium.  In  fact,  the  quantity 
of  work  immediately  succeeding  the  position  in  question  becomir.g 
negative,  shows  that  the  projection  of  the  virtual  velocity  is  negative, 
and  therefore  that  it  is  described  in  opposition  to  the  resultant  (jf  the 
forces,  which,  as  soon  as  it  overcomes  the  living  force  already  existing, 
will  cause  the  body  to  retrace  its  course. 


MECHANICS     OF    SOLIDS.  125 

§  1 3G. — If,  on  the  contrary,  the  body  reach  a  position  in  which  the 
quantity  of  work  is  a  nnnimum,  the  upper  sign  in  Equation  (124) 
must  be  taken,  the  second  member  will  always  be  positive  and  there 
will  be  no  limit  to  the  increase  of  V.  The  body  may  therefore 
depart  further  and  further  from  this  position,  however  small  V  may  be; 
and  hence,  this  is  called  a  position  of  unstable  equilibrium. 

§  137. — If  the  entire  second  member  of  Equation  (124),  be  zero, 
then  will, 

iJJ/F'2  -  1J/F2  =  0, 

and  there  will  be  neither  increase  nor  diminution  of  quantity  of  work, 
and  whatever  position  the  body  occupies  the  forces  will  be  in  equili- 
brio.     This  is   called   equiUbrium  of  indifference. 

§138. — If  the  system  consist  of  the  union  of  several  bodies  acted 
upon  only  by  the  force  of  gravity,  the  forces  become  the  weights 
of  the  bodies  which,  being  proportional  to  their  masses,  will  be  con- 
stant. Denoting  these  weights  by  IF',  TF",  IF'",  &c.,  and  assinn- 
ing  the  axis  of  z  vertical,  Ave  have  from   Equations  (87), 

Rz,  =  W'z'  +  W"z"  +  W"'z"'  +  &c., 

in  which  B,  is  the  weight  of  the  entire  system,  and  z^  the  co-ordi- 
nate of  its  centre  of  gravity;    and  differentiating, 

Rdz,  =  W'dz'  +  W"dz"  +  W"'dz"'  -f  &c.    .     .     .    (125) 

Now,  if  z^  be  a  maximum  or  minimum,  then  will 

W  dz'  +   W"  dz"  +   W'dz'"  +  &c.  =  0, 

which  is  the  condition  of  equilibrium  of  the  weights.  Whence,  wo 
conclude  that  when  the  centre  of  gravity  of  the  system  is  at  the 
highest  or  lowest  point,  the  system  will  be  in  equilibrio. 

In  ordei  that  the  virtual  moment  of  a  weight  may  be  positive, 
vertical  distances,  when  estimated  downwards,  must  be  regarded  as 
positive.  This  will  make  the  second  differential  of  z^ ,  positive  at 
the  limit  of  the  highest,  and  negative  at  the  limit  of  the  lowest 
point.  The  equilibrium  will,  therefore,  be  stable  when  the  centre  of 
gravity  is  at  the  lowest,  and  unstable  when  at  the  highest  point. 


12U  ELEMENTS     OF     ANALYTICAL     MECHANICS 

Integrating    Equation     (1-5),     between    the     limits     z^  =  11^    and 
z,  —  H\  z'  =  h^  and  z'  —  A',  &c.,  and  we  find, 

R{H^  -  II')  =  W'{h,  -  h')  +  W"  (A,,  -  A")  +  &c.  ;  .  (120) 

from  whieli  we  see  that  the  work  of  the  entire  weight  of  the  system, 
acting  at  its  centre  of  gravity,  is  equal  to  the  sum  of  the  quantities 
of  work  of  the  comjDonent  weights,  which  descend  diminished  by  the 
sum  of  the  quantities  of  work  of  those  which  ascend. 


INITIAL    CONDITIONS,    DIKECT    AND     INVERSE    PKOBLEM. 

§  130. — By  integrating  each  of  Equations  (120)  twice,  we  obtam 
three  equations  involving  four  variables,  viz.  :  .t,  y,  z  and  t.  By 
eliminating  t,  there  will  result  two  equations  between  the  variables 
X,  y  and  s,  which  will  be  the  equations  of  the  path  described  by 
the   centre   of  inertia  of  the   body. 

§  140. — In  the  course  of  integration,  six  arbitrary  constants  will 
be  introduced,  whose  values  are  determined  by  the  initial  circum- 
stances of  the  motion.  By  the  term  initial,  is  meant  the  ej^och 
from   which   t  is    estimated. 

The  initial  elements  are,  1st.  The  three  co-ordinatos  which  give 
the  position  of  the  centre  of  inertia  at  the  epoch ;  and  2d.  The 
component  velocities  in  the  direction  of  the  three  axes  at  the  sr.nie 
instant. 

The  general  integrals  determine  the  nature  only,  and  not  the 
dimensions    of  the   path.  ^ 

§141. — Now  two  distinct  propositions  may  arise.  Either  it  nia\ 
be  required  to  find  the  path  from  given  initial  conditions,  or  to 
find   the   initial  conditions   necessary    to   describe   a  given   path. 

fn  the  first  case,  by  integrating  Eqs.  (120)  twice,  we  obtain  six  eqiia- 

,     .  .        dx       dii       dz 
tions  in  .r,  y,  5',  /,  the  component  velocities,  — ,     -j- ,     -— ,  and  >ix  ■•ui)'.. 

traiy  constants  of  integration.     IMaking  in  these  equations    ^  =  0,    and 
substituting  for    the  co-ordinates    and  component  vi'locitics  ihoir  initial 


MECHANICS    OF    SOLIDS. 


127 


values,  the  constants  become  known.  These,  in  the  throe  equations 
obtained  from  bast  integration,  give  three  equations  in  .r,  y,  z  and  t,  from 
which,  if  t  be  eliminated,  two  equations  in  x,  y  and  z,  will  result.  These  will 
be  the  equations  of  the  path,  and  the  problem  will  be  completely  solved. 
In  the  second  case,  the  two  equations  of  the  path  being  differentiated 
twice  and  divided  each  time  hy  dt,  give  only  four  equations  involving 
tlirce  first,  and  three  second  differential  co-efficients.  The  inverse  problem 
is,  iherefore,  indeterminate. 

But   Equation    (121)  being    differentiated    and  divided   by  the   dif- 
ferential   of  one   of  the    variables,  say  dx,  gives 


dV^ 
iJ/  -— - 
^      ■  dx 


dx  dx 


(127) 


which  is  a  fifth  equation  involving  X,  Y,  Z,  and  V.  By  assuming 
a  value  for  any  one  of  these  four  quantities,  or  any  condition  con- 
necting them,  the  other    five    may  be  found  in  terms  of  .r,  y  and   z. 


VEETICAL   MOTION   OF   HEAVY   BODIES. 

§  142. — When  a  body  is  abandoned  to  itself,  it  falls  tOAvard  the 
earth's  surface.  To  find  the  circumstances  of  motion,  resume  Equa- 
tions (120),  in  which  the  only  force  acting,  neglecting  the  resistance 
of  the  air,  will  be  the  weight  =  Mg ;  and  we  shall  have,  Equa^ 
tions  (IIT), 

2  P  cos  a  =  J\r  =  Mcj  .  cos  a  ; 

2  P  cos  f^j  —  Y  —  Mg  .  cos  /3  ; 

2  P  cos  y  =  Z  =  Mg  .  cos  y ; 

in  which  M  denotes  the  mass  of 
the  body.  Tlic  force  of  gravity 
varies  inversely  as  the  square  of 
the  distance  from  the  centre  of 
the  earth,  but  within  moderate 
limits  may  be  considered  invaria- 
ble. The  weight  will  therefore  be 
constant  during  the  fall. 

Take  the  co-ordinate  z  vertical, 
and  positive  when  estimated  downwards,  then  will 

cos  «,  =  0  -,     C(^s  3  —  0  ;     cos  "'  --=  1, 


12S  ELEMENTS     OF    ANALYTICAL    MECHANICS. 

and  Equations  (120)  become,  after  omitting   the  common   factor  3f, 

d"x   __  (Py_  _  cP^  _ 

df'  ~      '     dfi   ~      '      dt^   ~^' 

and   integrating^ 

dx  dy 

dt        ^'  dt        y' 

^=  ^  =y^+«. (128) 

ill  which  V  is   the  actual  velocity  in   a  vertical   direction. 
Making  ^  =  0,  we   have 

dz  _ 
dTt  "  ^*^' 

The  constants  n  ,  u  and  u  ,  are  the  initial  velocities  in  the 
directions  of  the  axes  x,  y  and  2,  resiDcctively.  Supposing  the  first 
two  zero,  and   omitting   the   subscript   z,  from  the   third,  we   have, 

dx        ^      dii 

v='-^^  =  gt  +  u (129) 

Integrating  again,  we  find 

x=.  C;     y=  C\ 
z  =  irjf^  +  ut+  C", 

and  if   when    i  =  0,  the   body  be   on   the  axis   2,  and   at   a   distance 
below   the   origin    equal    to    «,    then    will 

X  =  Q;     y  =  0; 
z  =  lgi-  +  ut  +  a (130) 

If  the  body  had  been  moving  upwards  at  the  epoch,  then  would 
u   have   been   negative,  and,  Equations  (129)  and  (130), 

V  ziz  gt  —  n (131) 

z  =  Igfi  —  ut  +  a (132) 


MECHANICS     OF    SOLIDS.  129 

If  the  body  had  moved  from  rest  at  the  epoch  and  from  the 
orighi  of  co-ordinates,  then  would  v  be  the  actual  velocity  generated 
by  the  body's  weight,  and  z  =  h,  the  actual  space  described  m  the 
time  t;  and  Equations  (1^9)   and  (130)  would  become, 

V  -^  </i (133) 

k  =ic/t^ (134) 

and  eliminating  t, 

V  =  ^2ffh (135) 

whence,  we  see  that  the  velocity  varies  as  the  time  in  which  it  is 
generated ;  that  the  height  fallen  through  varies  as  the  square  of  the 
time  of  fall ;  and  that  the  velocity  varies  directly  as  the  square  root 
of  the  height. 

The  value  of  h,  is  called  the  height  due  to  the  velocity  v  ;  and 
the  value    v,  is  called  the   velocity  due  to  the  height  h. 

If,  in  Equation  (132),  we  suppose  a  =  0,  we  shall  have  the  case 
of  a  body  thrown  vertically  upwards  Avith  a  velocity  u,  from  the 
origin,  and  we  may  write, 

V  =  fft  —  V,     .     .  ■ (136) 

z  =  Ifft^  —  ui; (137) 

when  the  body  has  reached  its  highest  point,  v  will  be  zero,  and  we 
find, 

ff  t  —  u  =  0 ; 
or, 

u 
~   9  ' 

which  is  the  time  of  ascent;  and  this  value  of  ^,  in  Equation  (137), 
will  giv(i  the  greatest  height,  li  =  z^  to  which  the  body  will  attain, 

"  =  -|^ ('38) 

§  143. — In  the  preceding  discussion,  no  account  is  taken  of  the 
atmospheric  resistance.      For  the  same  body,  this  resistance  varies  as 

9 


130  ELEMENTS     OF     ANALYTICAL    MECHANICS. 

the  square  of  the  velocity,  so  that  if  k,  denote  the  velocity  when  the 
resistance  becomes  equal  to  the  "body's  weight,^  then  will 

F        ' 

be  the  resistance   when   the   velocity  is  v,  and  in  Equations  (117),  we 
shall  have, 


2  F  cos  a  =z  X  =.  M  g  cos  a.  +  Mg  •  —  •  cos  a', 


2  P  cos  /3  =  Y  =  M  g  cos^  -\-  Mg  •  —  •  cos  /3', 


2  P  cos  y  z=i  Z  =  M g  cos  y  +  M g  •  —  •  cos  y' ; 

f€ 


taking  the  co-ordinate  z,  vertical  and  positive  downward,  then  will, 

cos  a  =  cos  a'  =  0, 
cos  /3  =  cos  /3'  r=  0, 
cos  7  =  1,    cos  7'  =  —  1  ; 

and,  supposing  the  body  to  move  from  rest,  Equations  (120),  give, 

d^  z  v^ 

M-  -TV  =:  Mg  —  M q  •  — - 

Omitting  the  common  factor  M,  and  rcplacinir  — —  by  its  value  -r- . 


dv 
dTt 


whence, 


k'^.dv  h   /     dv       ,         dv     \  /,or.\ 


Integrating  and  supposing  the  initial  velocity  zero, 


gt  =  ^k. log    ^^ (140) 


MECHANICS     OF    SOLIDS.  131 

which  gives  the  time  in  terms  of  the  velocity;  or  reciprocally, 

Ic  +  V  -r 

in    which  e,  is    the  base  of  the    Naperian    system  of  logarithn'_s,  and 
from  which  we  find,  p  j^^  ^  J^ 

Ail 

V  =.  1 (142) 

11        -il 

•       which    gives   the  velocity    in    terms  of  the    time.     Substituting  for  v, 
—      its  value   —^5     integrating    and    supposing    the   initial    space   zero,    we 
have 

Qt  g  t 


Z   =: 


^•logi  ('e'^-    +  e    *^ (143) 


^^^^^         -^  -■  _r^^  V'^.^  ^-^     ^ 


130  ELEMENTS     OF    ANALYTICAL    MECHANICS. 

the  square  of  the  velocity,  so  that  if  /:,  denote  the  velocity  when  the 
resistance  becomes  equal  to  the  body's  weight,  then  will 


>  /c     .. 


M .  g  .v^ 


be  the  resistance  when   the   velocity  is  /;,  and  in  Equations  (IIT),  we 
shall  have, 


2  P  cos  a  =  X  =  if  ^  COS  a  +  A£ g  •  —  •  cos  a', 


2  P  cos  /3  —  Y  =  M  g  cos^  +  Mg  •  —  •  cos  /3', 

A/' 


2  P  cos  7  =:   Z  =  M g  cos  y  +  M g  •  —  •  cos  7' ; 
taking  the  co-ordinate  s,  vertical  and  positive  downward,  then  will. 


m 


MECHANICS     OF     SOLIDS.  131 

which  gives  the  time  in  terms  of  the  velocity;  or  reciprocally, 

^'  +  "   =  .    '"^ (141) 


k  —  V 


in    which  e,  is    the  base  of  the    Naperian    system  of  logarithms,  and 
from  which  we  find, 


V  =  


g_t 


(142) 


•       which    gives   the  velocity   in    terms  of  the    time.     Substituting  for  v, 

dz 
its  value   — i     integrating   and    supposing    the   initial    space   zero,    wf- 
dt 


have 

z  = 

12                       ^ ' 
-•logi  (e~^ 
9       "-  \ 

+ 

Multiplying 

Equation 

(139)   by 

dz 

we   have. 

9d^=   ,2 

dv 

9  t 


(143) 


ai  d   integrating,  observing   the   initial   conditions   as   above, 
which   gives    the   relation   between  the    space   and   velocity. 

_  £i 

As  the  time  increases,  the  quantity  e  ''  becomes  less  and  less, 
and  the  velocity,  Equation  (142),  becomes  more  nearly  uniform  ; 
for,  if  t  be   infinite,  then  will 

e     *  =  0, 
and,    Equation    (142), 

V  =z  k; 

making   the   resistance   of  the   air   equal   to   the   body's   weight. 


132  ELEMENTS     OF     ANALYTICAL     MECHANICS. 

§144. — If    the    body    had    been   movhig    upwards    with   a    velocity 
V,    then,  taking    z    positive    upwards,    would,    Equations    (120), 

substituting^   —     for   -^—7)   and  omitting  the  common  factor,  we  find, 
°  d  t  at" 


k .dv  g  d  t 


^.2  _^  ^3  ^.    ' 

integrating, 

and    supposing   the   initial  velocity  equal  to  a,  we  find 

-1  a 

C  =  tan    -7-5 


(145) 


ind,  / 


t/i2-. 


Taking  the   tangent   of  both  members  and  reducing,  we  find 

a  —  h  .  tan  '— 

v^lc ^^ (147) 

q  t 

]c  +  a.  tan  V 

k 

which  may  be  put  under  the  form, 

a  .  cos k .  sm  — 

v  =  k ^- f.       .     •     .     •     (148) 

a  .  sm  — — t  K  .  cos  — 
a;  k 

Substituting    for  v    its  value   -^^^     intogi-ating,    and    supposing    the 
initial  space  zero,  we  have 

= log  (  —  •  sm  •—  +  cos  —  )  •     •     •     •    (14y) 

o        °  \  k  k  k  / 


9 


4-  Jte^' 


MECHANICS     OF    SOLIDS.  133 

Multiplying  Equation  (145),  by 

dz 

and  we  have, 

h"^  .v.clv 


g  .dz  =  — 


A,-2  +  v^ 


and  integrating,  with  the  same  initial  conditions  of  v   being    equal    to 
a,  when  z  is  zero,  there  will  result, 

2  ^  il  .  log  i:l±i^- (150) 

§145. — If  we  denote  by  A,  the   greatest  height  to  \\hich  the  body 
will  ascend,  we  have  2  =  A,  when  v  ■=z  0,  and  hence, 

"  =  ^->-^^ (-) 

Finding  the  value  of  ;',  from  Equation  (14G),  we  have, 

«  =  —  (  tan tan      —)....     (152) 

q    \  k  k  J 


from  which,  by  making  ?;  =  0,  we  have, 

k           -1   a 
;    ^  A  .  tan      -f        (153) 

which  is  the  time  required  for  the  body  to  attain  the  greatest  eleva- 
tion. Having  attained  the  greatest  height,  the  body  will  descend,  and 
the  circumstances  of  the  fall  will  be  given  by  the  Equations  of  §  143. 
Denoting  by  a',  the  velocity  when  the  body  returns  to  the  point  (;f 
starting,  Equation  (144),  gives, 

^•2  k-' 

and  placing  this  value  of  h  equal  to  that  given  by  Equation  (151), 
there   will   result, 

^•2  __    F  -I-  a2 

yt2  -  a'2    ~  P       ' 


131  ELEMENTS     OF     ANALYTICAL    MECHANICS. 

whence, 

^•2 


a2  4-  X-2 


that    is,  the  velocity  of   the  body  %yhen    it   returns    to    the    point    of 
departure  is  less  than  that  with  which  it  set  out. 
Making  v  —  a'  in  Equation  (140),  we  have, 

k       ,       k  -\-  a' 

t  J.     =    •  lOff  i 

/  2^         ^   k  -  a'' 

and,  substituting  for  a',  its  value  above, 

t     =-—-.  log  "^  ,  .     .     .     .       154) 

a  value  very  different  from  that  of  jf„,   given  by  Equation  (153),  for 
the  ascent. 

Multiplying  both  numerator  and  denominator  of  the  quantity  whose 
logarithm  is  taken,  by   -yj  a?  -{-  k'^  —  a,    the  above  becomes, 

A-dding  Equations  (153)  and  (155),  we  have. 


^'  r    -1  «    ,  ,  k         -| 

—    tan      -; — h  los 

^   L  k  '^  ^  a^  ^  ]c-i  -  aA 


^        ff   I  k  "  y  a2  +  A;2 

0-,  making  t  =  t^  +  t^, 

If  a  ball  be  thrown  vertically  upwards,  and  the  time  of  its 
absence  from  the  surface  of  the  earth  be  carefully  noted,  t  will  be 
known,  and  the  value  of  k  may  be  found  from  this  equation.  '^Ihis 
experiment  being  repeated  with  balls  of  diliercnt  diameters,  and  the 
resulting  values  of  k  calculated,  the  resistance  of  the  air,  for  any 
given  velocity,  will  be  known. 


MECHAN'ICS     OF     SOLIDS. 


135 


PEOJECTILES. 

§  146. — Any  body  projected  or  impelled  forward,  is  called  a  pro- 
jectile^ and  the  curve  described  by  its  centre  of  inertia,  is  called  a 
frajectonj.  The  projectiles  of  artillery,  which  are  usually  thro-wn  with 
great  velocity,  will  be  here  discussed. 

§  147. — And  first,  let  us  consider  what  the  trajectory  would  be 
in  the  absence  of  the  atmosphere.  In  this  case,  the  only  force  which 
acts  upon  the  projectile  after  it  leaves  the  cannon,  is  its  own  weight ; 
and.  Equations  (IIT), 

2  P  cos  a.  =!  X  z=  M g  cos  a, 
2  P  cos  /3  =  Y  =  Mg  cos  ,8, 
2  P  cos  7  =   Z  =  Mg  cos  y. 


Assumins;  the  orijjin 
at  the  point  of  de- 
parture, or  the  mouth 
of  the  piece,  and 
taking  the  axis  z 
vertical,  and  posi- 
ijye  upwards,  then 
will 


cos  a  =  0  ;    cos  ,<3  =  0;    cos  7  =  —  1  ;    and.  Equations  (120), 


M-  -7^  =  0 


M- 


d  fi 


'■'  ''■%  =  -^''-' 


and  integrating,  omitting  i/. 


d  X 

~dl 


d  y 
dT 


d_ 
~dt 


,1*  It    ^  ^1     4-  ^  '  5 


(157) 


Integrating  again,  and  recollecting  that  the  initiiil  spaces  are  zero,  Ave 
have. 


^  =^  «.  •  ^ ;   y 


V  •  t 

y 


hO^-''  +  «    •  ^ 


(158) 


lyO  ELEMENTS     OP    ANALYTICAL    MECHANICS 

and  eliminating  t,  from  the  first  two,  we  obtain, 


u 

y 
y  =  —  .  x; 


which    is    the    equation  of  a   right    line,  and    from  which  we  see  that 

the  trajectory  is  a  plane  curv^e,  and  that  its  plane  is  vertical. 

Assume  the  plane  zx,  in  this  plane,  then  will  y  =  0,  and  Ecpia- 
tions  (158),  become, 

X  =  u^-  t;    z  =-.  —  l-fft^  +  u^'  t.      .     .     .      (159) 

Denote  by  V,  the  velocity  with  which  the  ball  leaves  the  piece, 
that  is,  the  initial  velocity,  and  by  a,  the  angle  which  the  axis  of  the 
piece  makes  with  the  axis  x,  then  will, 

V.  cos  a,    and     V  .  sin  a, 

be  the  lengths  of  the  paths  described  in  a  unit  of  time,  in  the  direc- 
tion of  the  axes  x  and  z,  respectively,  in  virtue  of  the  velocity  V  ; 
they  are,  therefore,  the  initial  velocities  in  the  directions  of  these 
axes ;    and  we  have, 

M    =  F  cos  a  ;     ?(    =  V.  sin  a  ; 
which,  in  Equations  (159),  give 

a;=F.  cos  a.  ^;     z  ^  -  ^  r/ 1^  +  V  .sin  a  .t      •     .     (160) 
and   eliminating  t,  we  find 

z  =  X  tan  a 


2  F2 .  cos2  a  ' 
or   substituting  for    V  its  value    in  Equation  (135), 


=z  X  tan  a  —  -r-r-^ — r— (1^1) 

4  h  .  COS"*  a 


which  is  the  equation  of  a  parabola. 


MECHANICS     OF    SOLIDS. 


137 


§  148. — The  angle  a  is 
called  the  angle  of  projec- 
tion ;  and  the  horizontal 
distance  A  D,  from  the 
place  of  departure  A^  to 
the  point  Z),  at  which  the 
projectile  attains  the  same 
level,  is  called  the  range. 

To  find  the  range,  make  2  =  0,  and  Equation  (101)  gives 

a;  =:  0,    and    a;  =  4  A  sin  a  cos  a  =  2h  sin  2 a, 

and    denoting   the   range   by  B, 

E  =  2h.sm2a (102) 

the    value    of  which   becomes    the   greatest    possible    when   the   angle 
of   projection   is   45°.     Making  a  =  45°,  we   have 


B  =  2h 


(103) 


that   is,    the    maximum     range    is    equal    to    twice    the   height  due   to 
the   velocity  of  projection. 

From  the  expression  for  its  value,  we  also  see  that  the  same 
range  will  result  from  two  different  angles  of  projection,  one  of  which 
is    the    complement  of  the    other. 

§  149. — Denoting  by  v  the  velocity  at  the  end  of  any  time  t,  we 
have, 

2         d  s^        dz^  +  d  x^ 
~  ~dfi  ~         'd'tP' 

or,  replacing  the  values  of  dz  and  d  x,  obtained  from  Equations  (100), 

V-  =  V^  —  2  V.g.t.  sin  a  +  g^i^    ....     (1G4) 

and   eliminating   t,    by    means    of  the    first    of  Equations    (100),    and 
replacing   V^,  in    the  last  term   by  its  value  2gh, 


y2  —.  p^2  _  2  (7 .  tan  u  .  X  +  c  • 


z  ii .  cos^  a 


(105) 


138  ELEMENTS     OF     ANALYTICAL    MECHANICS. 

in  which,  if  we   make   x  =  Ah.  sin  a  cos  a,  we   have   the   velocity   at 
the   point  i>, 

«hich  shows  that  the  velocity  at  the  furthest  extremity  of  the  range 
is    equal    to    the   initial    velocity. 

Differentiating  Equation  (161),  we  get 

^^  =  taM  =  tan  a  -  j^-j-  .     .     .     .     (166) 

in  which   ^   is   the   angle  which   the   direction  of  the   motion   at  any 
instant  makes   with   the  axis   x. 
Making   tan  ^  =  0,  we  find 

a;  =  2  A  ,  cos  a  .  sin  a, 

which,  in  Equation    (161),  gives 

z  =  h.  sin^  a, 

the  elevation  of  the   highest  point. 

Substituting  for  x,  the  range,  4  h  cos  a  sin  a,  in   Equation  (166), 

tan  ^  —  —  tan  a, 

which    shows   that   the   angle  of  fall  is  equal  to  minus  the  angle  of 
projection. 

g  150. — The  initial  velocity  V  being  given,  let  it  be  required  to 
find  the  angle  of  projection  which  will  cause  the  trajectory  to  pass 
through    a   given  point  whose   coordinates  are   x  —  a  and   z  =z  b. 

Substituting   these   in   Equation    (161),  we   have 

b  =  a  tan  a 


4  h  .  COS" a 


from   which   to   determine   a. 
Making   tan  a  =  cp,  we   find 


1  + 


MECHANICS     OF    SOLIDS. 


139 


wliich   in  the   equation   above,    gives 
4  /i .  i  +  cr  —  4h.a. 


+  a- 


whence, 


tan  a 


2  A 
a 


-v/IP 


4hb  —a^  . 


(167) 


The  double  sign  shows  that  the  object  is  attained  by  two  angles, 
and  the  radical  shows  that  the  solution  of  the  problem  will  be 
possible   as   long   as 

4  /i2  >  4  A  5  4-  a\ 

Making, 

4  /i2  -  4  A  .  6  —  a2  z=  0, 

the  question  may  be  solved  with  only  a  single  angle  of  projection. 
But  the  above  equation  is  that  of  a  parabola  whose  co-ordinates  are 
a  and  b,  and  this  curve  being  con- 
structed and  revolved  about  its  vertical 
axis,  will  enclose  the  entire  space 
within  which  the  given  point  must  be 
s-ituated  in  order  that  it  may  be  struck 
with  the  given  initial  velocity.  This 
[)arabola  will  pass  through  the  farthest 
extremity  of  the  maximum  range,  and 
at  a  height  above  the  piece  equal  to  h. 

§  151.- — Thus  we  see  that  the  theory  of  the  motion  of  projectiles 
is  a  very  siiiiplc  matter  as  long  as  the  motion  takes  place  in  vacuo. 
But  in  practice  this  is  never  the  case,  and  Avhere  the  velocity  is  "con 
siderable,  the  atmospheric  resistance  changes  the  nature  of  the  tra- 
jectoi'y,   and  gives  to  the  subject  no  little  complexity. 

Denote,  as  before,  the  velocity  of  the  projectile  when  the  atmos- 
pheric resistance  equals  its  weight,  by  k,  and  assuming  that  the 
resistance  varies  as  the  square  of  the  velocity,  the  actual  resistance 
at  any  instant  when  the  velocity  is  v,  will  be, 


—  31  cv^, 


140  ELEMENTS     OF     ANALYTICAL    MECHANICS, 

by  making, 

g 
¥  =  '■ 

The  forces  acting  upon  the  projectile  after  it  leaves  the  piece 
being  its  weight  and  the  atmospheric  resistance,  Equations  (120), 
become, 

d  t" 

M-''^  =  Mg.  COS  [3 -\- 3Ic.v^.  COS  13', 
d  fi 

d*^  z 
M  •  — r  =  ^f  0  •  cos  y  +  Mc .  v" .  cos  y' 

d  i'^ 

Taking  the  co-orclinates    z    vertical,    and    positive    when    estimated 
upwards, 

cos  a  ::=  0  ;     cos  /3  z=  0  ;     cos  y  =  —  1 , 

and  because  the  resistance  takes  place  in  the  direction  of  the  trajec- 
tory, and  in  opposition  to  the  motion,  if  the  projectile  be  thrown  in 
the  first  angle,   the  angles  c/,  /3',  and  7',    Avill  be  obtuse, 

dx  r,,  '^y  ,  ^^ 

cos  a'  — p-  ;     cos  /3'  = ;     cos  7'  = — , 

ds  ds  d  s 

and  the  equations  of  motion  become,  after  omitting  the  common 
factor  M, 

d^  X  _  2     ^  ^  . 

— 7 — T   — -  C  •  V    •       -       , 

d  i-  di-- 

T^-  "'■"  ■  d.  ' 

d^z  __  _      _         2    A^. 
'd¥~        ^       ^'^'"  ds' 


From  the  first  two  we  have,  by  division, 

C?2  y  d?'  X 

dy  d  X 


MECHANICS    OF    SOLIDS.  lil 

and  by  integration, 

log  dy  ^:z  log  dx  -\-  log  C; 

and,  passing  to  the  quantities, 

dy  ^=:  C  d  X. 

Integrating  again,  we  have, 

y  =  Cx-\-   C; 

in  which,  if  the  projectile  be  thrown  from  the  origin,  C"  =  0,  thus 
giving  an  equation  of  a  right  line  through  the  origin.  Whence  we 
see  that  the  trajectory  is  a  plane  curve,  and  that  its  plane  is  vertical 
through  the  j^oint  of  departure.  -  ~f 

Assuming    the   plane   z  x,  to    coincide   with   that  of  the   trajectory, 
and  replacing  v"^,  by  its  value  from  the  relation, 

d^    _       3 
d  fi    -    "  ' 

we  have, 


d'^  x                     d  s 

d  X 

dt^  ~        ^'     dt 

dt 

J 

d^2 

ds 

dz 

de-        '        " 

dt 

dt 

(168) 


From  the  first  we  have. 


d"^  X 

dt^  ds 


d  X  dt 


dt 


and  hj  integration, 


log     — —  =  —  c  .  s  +  U. 

"^       dt 


Denotuig  by  e,  the    base  of  the    Naperian    system    of    logarithms, 
and  making   C  =  log  A,  the  above  may  be  written, 

dx 


Ii2  ELEMENTS     OF    ANALYTICAL    MECHANICS, 

and  passing  from  logarithms  to  the  quantities, 

ii=^-   069) 

Denoting  by    V,  the  initial   velocity,  and   b}   a,  the   angle  of  pro- 
jection, we  have,  by  making  s  =  0, 

—r-  =  A  =:  V  cos  a, 

which   substituted   above,  gives 

^^'  =  F.cosa.e"^* (170) 

To   integrate  the  second   of  Equations  (168),  make 

dz  dx  /i^,x 

7^  =  i'-Ti ("1) 

in  which  p  is    an  additional    unknown    quantity. 

Differentiating    this    equation,     dividing     by    dt,     and     eliminating 

from   the   result,  -y-y'    ^^y    its    value   in    the   first  of  equations  (108), 
we   have, 

d'^z         dp     dx  ds  dx 

'dfi   ~  dTt  '  dTt  ~  ^^ '    '  I't'Ti 

and   substituting   this   value   in    the    second   of  Equations   (168),    we 
have,  after  eliminating    — ^  by  its  value,  obtained  from  Equation  (171), 

dx     dp 

dt     dt  "^  ^       ' 

and   dividing   this  by  the  square   of  Equation  (170), 


dp 

d± il 

dx  ~~  F-  cos^  a 

Tt 


(173) 


MECHANICS     OF    SOLIDS.  143 

but  regarding  s   and  2^  ^s  functions  of  x,  we  have,  Equation  (171), 

dz 

di 

and, 

dp 

d  i         dp 

d  X         d  X 

'dT 

whence,  making   F^  =  2c/h,  Equation  (173)  becomes 

2cs 


dp 


(175) 


d  X  2  A  .  cos^  a 

and  multiplying  this  by  the  identical  equation,  /    / 

obtained  from  Equation  (174),  we  find,  "    i.  , 

and   integrating, 

2cs 

in  which  C  is  the  constant  of  integration ;  to  determine  which,  make 
s  =  0  ;  this  gives  p  =  tan  a ;  and 


C  =  — -, —  -i-  tan  a  .  J\  +  tan^  a.  +  log  (tana  -r  -i/TTtan^ct)  •  (177) 

2  c  /i  cos^  a        t  ^  "  ^ 

From    Equation  (175)  we   have, 

—  2  c  s 

c?a;  =  —  2  A.  cos^  a  .  e         •  dp  ; 

from   Equation   (171), 

dz  =.  p  .  dx  ; 


14-i         ELEMENTS     OF     ANALYTICAL    MECHANICS. 

from  Ecjuation   (1'72), 

g  d  t"  ^z  —  d  X .  dp  ; 

and   eliminating   the  exponential  factor  by  means  gf  Equation.   (176); 
we  find, 

c.dx  =  '^ ;   .    (178) 

p  -y/l  -V'f  +  log  {p  +  yTTT")  -  ^ 

c.dz^ ^^^ ;.     (179) 

P  -/ 1  +  p"  +  log  (p  +  vTTi^)  -  G 

■y/T^  .dt  =  ~  '^^  ;  .    (180) 

J  G  -p  V  1  +  p"-  -  log  {p  +  VTTi^) 

Of  the  double  sign  due  to  the  radical  of  the  last  equation,  the 
negative  is  taken  because  p,  which  is  the  tangent  of  the  angle  made 
by  any  element  of  the  curve  ■with  the  axis  of  ar,  is  a  decreasing 
function  of  the  time  t. 

These  equations  cannot  be  integrated  under  a  finite  form.  But 
the  trajectory  may  be  constructed  by  means  of  auxiliary  curves  of 
which  (178)  and  (179)  are  the  differential  equations.  From  the  first. 
we  have, 

dx  ^  T  .dp; (181) 


and  from  the  second, 

dz  =  T. 

in  which, 

T-'  . 

1 

.    .    .     (182) 


;-(183) 


"     p  -v/TTi^+  log  ( V  +  ^/T\~f)  -  c 
and  dividing  Equations  (181)  and    (182),  by  dp), 

4^  =  T; (184) 

dp 


MECHANICS     OF    SOLIDS. 


145 


Now,  regarding  x,  p^  and  2,  />,  as  the  variable  co-ordinates  of  two 
auxiliary  curves,  T,  and  T .  p,  will  be  the  tangents  of  the  angles 
which  the  elements  of  these  curves  make  with  the  axis  of  p. 

Any  assumed  value  of  ^),  being  substituted  in  T,  Equation  (183), 
will  give  the  tangent  of  this  angle,  and  this.  Equation  (184),  multi- 
plied by  dj),  will  give  the  ditlerence  of  distances  of  the  ends  of  the 
corresponding  element  of  the  curve  from  the  axis  of  ^;,  Beginning 
therefore,  ^t  the  point  in  which  the  auxiliary  curves  cut  the  axis  of 
2),  and  adding  these  successive  differences  together,  a  series  of  ordi- 
nates  x  and  2,  separated  by  intervals  equal  to  d]),  may  be  found,  and 
the  curves  traced  through  their    extremities. 

At  the  point  from 
which  the  projectile 
is  thrown,  we  have, 

a:  =  0  ;  z  —  0  ;  j:)==:tan  «, 

and  the  auxiliary 
curves  will  cut  the 
axis  of  p,  in  the  same 
point,  and  at  a  dis- 
tance from  the  origin  equal  to  tan  a.  Let  A  B,  be  the  axis  of  p, 
and  AC,  the  axis  of  x  and  of  2;  take  AB  =  tan  a,  and  let  BzJ), 
and  BxE,  be  constructed  as  above. 

Draw    the    axes  Ax   and   Az,  througb   the   point  of  departure  ^1, 
Fig.    (S)  ;     draw    any 
ordinate  c  z,  x^  to  the 
auxiliary    curves    Fig. 
(1);  lay   off  Ax,Y\g.   ^ 
(2)  equal  to  ^x,  Fig.*, 
(1),  and  draw  through 
x, ,     the     line     x^     z^ 
parallel    to    the    axis 
A  z,  and  equal  to  c  z. 
Fig.    (1)  ;     the    point 
Zi  will    be  a  point    of 

the  trajectory.     The  range  AD.  is  equal  to  ED,  Fig.  (1). 

10 


14:6  ELEMENTS     OF    ANALYTICAL    MECHANICS. 

By  reference  to  the  value  of  C,  Equation  (177),  it  will  be  seen 
that  the  value  of  T,  Equation  (183),  will  always  be  negative,  and 
that  the  auxiliary  curve  whose  ordinates  give  the  values  of  x,  can, 
therefore,  never  approach  the  axis  of  ^;.  As  long  as  ^;  is  positive, 
the  auxiliary  curve  whose  ordinates  are  2-,  will  recede  from  the 
axis  p ;,  but  when  j?  becomes  negative,  as  it  will  to  the  left  of 
the  axis  A  C,  Fig.  (1),  the  tangent  of  the  angle  which  the  element 
of  the  curve  makes  with  the  axis  p,  will,  Equation  (185),  become 
positive,  and  this  curve  will  approach  the  axis  p,  and  intersect  it  at 
some   point  as  D. 

The  value  of  -p  will  continue  to  increase  indefinitely  to  the  left 
of  the  origin  yl.  Fig.  (1),  and  when  it  becomes  exceedingly  gi-eat, 
the  logarithmic  term  as  well  as  (7,  and  unity  may  be  neglected  in 
comparison  with  jj,  which  will  reduce  Equations  (178)   and  (1*79)   to 


dx   =  -^; 

dz  =       ^ 
c  .p 

X  =  C  -  —;     z 
cp 

^  C"  +  - 
c 

and   integrating, 


which  will  become,  on  making  p  very  great. 


log  i', 


X  z^  C  ;     z  =  C"  +  -  log  ;?, 

which  shows  that  the  curve  whose  ordinates  are  the  values  of  x, 
will  ultimately  become  parallel  to  the  axis  p,  while  the  other  has 
no  limit  to  its  retrocession  from  this  axis.  Whence  we  conclude, 
that  the  descending  branch  of  the  trajectory  approaches  more  and 
more  to  a  vertical  direction,  which  it  ultimately  attains ;  and  that 
a  line  6rZ,  Fig.  (2),  perpendicular  to  the  axis  a:,  and  at  a  distance 
from  the  point  of  departure  equal  to  C",  will  ])e  an  asymptote  to 
the  trajectory. 

This  curve  is  not,  like  the  parabolic  trajectory,  symmetrical  in 
reference  to  a  vertical  through  the  highest  point  of  the  curve ; 
the  angles  of  falling  will  exceed  the  corresponding  angles  of  rising, 
the  range  will  be  less  than  double  the  abscissa  of  the  highest  point, 
and  the  angle  which  gives   the  greatest   range  will  be  less  than  45°. 


MECHANICS     OF    SOLIDS. 


H7 


Denoting    the    velocity    at   any    instant  by    v,   we    have 
dx^  -\-  ds^        ,,  „.    d  x" 

and  replacing    dx^    and    d  t'^   by  their  values    in  Equations  (178)  and 
(180),  Ave  find 

2         1  r/.(l+^2) 


C  -J)  V  1  +2^  -  log  {P  +  /I  +  p'') 


(186) 


and  supposing  p  to  attain  its  greatest  value,  which  supposes  the 
projectile  to  be  moving  on  the  vertical  portion  of  the  trajectory, 
this  equation  reduces,  for  the   reasons   before   stated,  to 


v  =  y/- 


which  shows    that    the    final   motion  is  uniform,  and  that  the  velocity 

will    ther    be    the    same    as    that    of  a    heavy    body    which  has  fallen 

1  P 

in  vacuo  through  a  vertical  distance  equal  to — 

'^  -^  2c  2(7 

§  152. — When  the  angle  of  projection  is  very  small,  the  projectile 
rises  but  a  short  distance  above  the  line  of  the  range,  and  the  equation 
of  so  much  of  the  trajec- 


Z 


tory  as  lies  in  the  imme- 
diate neighborhood  of  this 
line  may  easily  be  found. 
For,  the  angle  of  projec- 
tion being  very  small,  p 
will  be  small,  and  its 
second  power  maj^'  be 
neglected  in  comparison 
with  uiiity,  and  we  may 
take, 

d  s  =.  d  X  ;    and    s  r=  .r ; 

which  in  Equation  (175),  gives. 


J) 


(Ly. 


~  -  e. 


^t3 


^hc^^^ 


dp 
dx 


d-^z 
dx^ 


2  h .  cos^  a 


(187) 


^  V,. 


^ity.^)/h(yef)  ^  ^ 


L4S  ELEMENTS     OF    ANALYTICAL    MECHANICS. 


Integrating, 


d  z 


dx  4c.h.cos^a 


makiLg  X  =  0,  we  nave  — ; —  =  tan  a, 
^  '  dx  ' 


+  O; 


whence. 


O  =  tan  a  + 


1 


4  c  .  A  .  cos^  a 


which  substituted  above,  gives 
dz 


— —  =  tan  a  — f • 

d  X  4c  .  h  .  cos2  a       4c  .  h  .  cos^  a 


and  integrating  again 


z  =  tan  a  .  a; 


+ 


c^  .  h  ,  cos^  a       4c .  h  .  cos^  k 
making  .t  =  0,  then  will  2  =  0,  and 


+  c\ 


c  = 


8c^  .  h.  cos^  a ' 


hence, 


2  =  tana.r—  — -— ~{e      —2cx  —  l).     .     (188) 


From  Equation  (172),  we  have, 

g  .d  fi  z=i  —  dx  .  d]), 
and  substituting  the  value  of  d ]),  from  Equation  (187), 


dt  = 


e      .   dx 


y2c/h.  cos  a ' 
and  integrating,  making  x  =  0,  when  t  =  0, 


t  =z 


c  -y/  2ff  h  .  cos  a 


(Z'-l)     ....      (ISO) 


AiECHANlOS     OF    SOLIDS.    •  149 

M'hich   will    give    the    time    of    flight    to    any    point   whose   hoi'izonttiJ 
distance  from   the  piece   is   equal   to   x. 

§  153. — Let  the  projectile  fall ,  to  tlie  ground  at  the  point  I),  and 
denote  the  co-ordinates  of  this  point  by  x  =  /,  and  z  =  X,  and  sup- 
j)ose  the  time  of  flight  or  t  ~  r.  These  values  in  Equations  (188) 
and    (189),  give 

—  Sc^.h.  cos^  a  (X  —  I .  tan  a)r=e^— 2c^—  1    •    (189)' 


cos  a.  .T  .  c  .  -^/^yh  —  e'^     —  1      ....  (1S9)" 

When  the  two  constants  h  and  c,  as  well  as  a  and  X,  are  known, 
these  equations  will  give  the  horizontal  distance  Z,  and  the  time  of 
flight.  Conversely,  when  the  quantities  a,  ?,  X  and  t  are  known, 
they  give  the  co-efficient  of  resistance  c,  and  the  height  h,  due  to 
the  velocity  of  projection,  and  therefore,  Equation  (135),  the  initial 
velocity  itself. 

Eliminating  the  height  h,  we  find 

-  4  (X  -  lAana){c'^  -  1)2  =  ^  .  t2  .  (/''  -  2  c  I  -  1)  ■  •  (180)'" 

from  which   the  value  of  c  may  be  found,  and    one  of  the  preceding 
equations   will    give  A,  or   the  initial  velocity. 

it  may  be  worth  while  to  remark  that  if  the  exponential  term 
in  Equation  (188)  be  developed,  and  c  be  made  equal  to  zero,  which 
is  equivalent  to  supposing  the  projectile  in  vacuo,  we  obtain  Equa- 
tion (161). 

§  154. — Assuming  that  the  resistance  of  the  air  varies  as  the  square 
of  the  velocity,  some  idea  may  be  formed  of  its  actual  intensity  from 
the  fact  that  a  twenty-four-pound  ball  projected  with  a  velocity  of  2,000 
feet  in  vacuo,  and  under  an  angle  of  45°,  would  have  a  range  of 
125,000  feet;  whereas  actual  experiment  in  the  air  shows  it  to  be  but 
VjSOO  feet — about  one-seventeenth  of  the  former. 

JNIany  circumstances  qualify  botli  the  patli  and  vclocitv  of  projectiles. 
The  law  of  the  resistance  may  be  the  sjime  ibr  all  figures,  but  it  is 
known,  from  actual  trial,  not  to  be  that  of  the  square  of  the  velocity, 
except  for  very  small  lat.'s  of  motion.      For  the  same  velocity,  the  in- 


1 50  £  L  E  il  E  ]S:  T  S     OF     A  N  A  L  Y  T I  C  A  L     M  E  C  II  A  X  I  C  S . 

tensity  of  the  resistance  varies  with  the  size  and  figure  of  tlio  i)all. 
Much  depends  upon  the  facihty  Avith  which  the  compressed  air  iu  iiorit 
may  escape  lattei'ly  and  make  its  way  to  the  I'car.  The  actual  resist- 
ance at  any  instant  is  composed  of  two  terms,  the  one  due  to  the 
inertia  of  the  dispUxced  particles,  the  other  to  the  diiferenc  of  at- 
mospheric pressure,  as  such,  in  front  and  rear.  If  during  the  Jiotion 
the  air  could  close  in  behind  and  exert  the  same  pressure  as  in  front, 
tlio  resistance  would  be  wholly  due  to  inertia.  If  the  ball  were  at  rest, 
and  all  the  air  removed  in  rear  of  the  plane  of  largest  section  perpen- 
dicular to  the  trajectory,  the  resistance  would  be  due  entirely  to  the 
barometric  pressure  on  the  extent  of  this  section.  Both  terms  of  the 
resistance  must  be  variable  and  a  function  of  the  velocity,  till  the  latter 
is  so  great  as  to  leave  a  vacuum  behind,  w^hen  the  barometric  term 
would  become  constant. 

From  a  careful  and  elaborate  investigation  of  the  numerous  experi- 
ments upon  this  subject,  Coh  Piobert  has  constructed  this  empirical 
formula  for  spherical  projectiles,  viz. : 


A  .7T  .  r 


(:+^),.  .:...(  , 


iu  which  p  is  the  resistance  in  kilogrammes,  v  the  velocity,  tt  the  ratio 
of  the  diameter  to  the  circumference,  r  the  radius  of  the  ball,  A  the 
resistance  on  a  square  metre  when  the  velocity  is  one  n  etre^  and  v^  the 
velocity  which  would  make  the  resistance  measured  by  the  second  term 
equal  to  that  measured  by  the  first. 

§  155. — If  the  ball  be  not  jjcrfectly  homogeneous  in  density,  the 
centre  of  inertia  will,  in  general,  be  removed  from  that  of  figure ;  the 
resultant  of  the  expansive  action  of  the  powder  will  pass  through  the 
latter  centre  and  communicate  to  the  ball  a  rotary  motion  about  tlie 
former.  The  atmospheric  resistance  will  be  greater  on  the  side  of  tlie 
greatest  velocity,  and  deflect  the  projectile  to  the  opposite  side. 


MECHANICS     OF     SOLIDS.  151 


EOTAKY   MOTIOISr. 

§  156. — Having  discussed  the  motion  of  translation  of  a  sinsjle 
body,  we  now  come  to  its  motion  of  rotation.  To  find  the  circum- 
stances of  a  body's  rotary  motion,  it  will  be  convenient  to  transform 
Equations  (118)  fi-om  rectangular  to  polar  co-ordinates.  But  before 
doing  this,  let  us  premise  that  the  angular  velocity  of  a  body  is  the 
rate  of  its  rotation  about  a  centre.  The  angular  velocity  is  measured 
by  the '  absolute  velocity  of  a  point  at  the  uniCs  distance  from 
the  centre,  and  taken  in  such  position  as  to  make  that  velocity  a 
maximum. 

§  157. — Both  members  of  Equations  (38)  being  divided  by  d  t, 
give 

d  x'  ,      e?  4*  f      (^  ^ 

dt     ~  ^    '    dt     ~   ^^    '  d  t 


d  y'  ^      d  (p  I      d  -m 

dt  '    d  t  '    d  t 

d  z'  ^      d  -ui  ,     f?  4^ 

'dT  ~  ^  ■  'd't  *  ■  ^y 


(190) 


in  which  the  first  members  taken  in  order,  are  the  velocities  of  any 
element,  as  .m.  in  the  direction  of  the  axes  .r,  y,  z,  respectively,  tn 
reference  to  fi  centre  of  inertia,  §  75,  while 

dzi       d-\^       d(p 

— z >       — ; — J       — — ■> 

at         dt       dt 

are  the   angular  velocities   about   the   same  axes  respectively. 

Denoting  the  first  of  these  by  v_,,  the  second  by  v^,  and  the  third 
by  Vj ,  we  have 

dn^  d-\>  da 


152 


ELEMENTS  OF  ANALYTICAL  MECHANICS 


and  Equations  (190) 

may  be  written 

dx'         ,               ,         . 

dt                '              "' 

dz' 

dt  -y-'^  •^■•vj 

(102) 


§  158. — If  an  element  m  be  so  situated  that  its  velocity  shall  be 
equal  and  parallel  to  that  of  the  centre  of  inertia,  then,  .for  this 
element,  will  each  of  the  first  members  of  Equations  (192)  reduce 
to  zero,  and 

2' .  v„  —  y' .  V,  =  0,     ^ 


s'  .  V,  =  0, 

x'.v^^O- 


(193) 


the  last  being  but  a  consequence  of  the  two  others,  these  equations 
are  those  of  a  right  line  passing  through  the  centre  of  inertia, 
every  point  of  \\hich  will  have  a  simple  motion  of  translation 
parallel  and  equal  to  that  of  the  centre  of  inertia.  The  Avhole 
body  must,  for  the  instant,  rotate  about  this  line,  and  it  is,  there- 
fore, called   the  Axis  of  Iiistuntaneuus  Rutatioa. 

§159. — Denote  by  a^ , 
/3^ ,  y^ ,  the  angles  which 
this  axis  makes  with  the 
CO  ordinate  axes  a;,  y,  0, 
respectively.  Then,  tak- 
ing any  point  on  the  in- 
stantaneous axis,  will. 


-X 


cos  (3^  = 


COS  7^ 


v^- 

y' 

+ 

2 '2 

^/^ 

+  y'^ 

+ 

?2 

V^-'2  -f  y'-^  +  z'2 


MECHANICS     OF     SOLIDS, 
and  eliminating  a;',  y'  and  z\  by  Equations  (193), 


153 


V\^  +  V  +  \ 


COS  (3^  = 


V'x"  +  "i/  +  "z 


COS  7^  = 


(194) 


V'x"  +  %^  +  ^'-^ 


which  will  give    the    position    of    the   instantaneous   axis    as    soon  as 
the   angular  velocities  about   the  axes  are   known. 

§  160. — Squaring    each  of   Equations    (192),  taking   their   sum    and 
extracting  square  root,  we  find 


y 


cix'-^  +  dy''^  +  dz'^ 


dfi 


=  ^::.V(.'.v^-y'.vJ"-  +  (..'.v-.X)2  +  (y'.v^-..V^; 


Replacing  v^,  v     and  v,  by  their  values  obtained  by  siniiply  clearing 
the    fractions   in    Equations    (194),    this    becomes 


V  =  v/v/  +  V^2  +  v,2  X  -/^^  +  y'2  -(-  £-'2  _  (^.f'  COS  OC,  +  y'  cos  /3^  +  Z'  COS  7J2^ 

wliich  is  the  velocity  of  any    element   in    reference    to   the  centre   of 
inertia. 
Making 

^'2    +    y'2    +    g'2    ^    1^ 

we  have  the  element  at  a  unit's  distance  from  the  centre  of  inertia  j' 
and  making 


x'  cos  a^  +  y'  cos  [3^  +  z'  cos  y^  =  0, 


(19o) 


the  point  takes  the  position,  giving  the  maximum  velocity.  In  this 
case  V  becomes  the  angular  velocity,  and  we  have,  denoting  the 
latter   by  v., 


^i  =  vV  +  ^'  +  ^ 


(IDG) 


loi 


ELEMENTS     OF    ANALYTICAL    MECHANICS. 


Equation  (195)  is  that  of  a  plane  passing  through  the  centre 
of  inertia,  and  perpendicuhir  to  the  instantaneous  axis.  The  position 
of  the  co-ordinate  axes  being  arbitrary,  Equation  (196)  shows  that 
the  sum  of  the  squares  of  the  angular  velocities  about  the  three 
co-ordinate  axes  is  a  constant  quantity,  and  equal  to  the  square  of 
the  angular  velocity  about   the   instantaneous   axis. 

§  161.— Multiply  Equation  (196),  by  the  first  of  Equations  (194), 
find  there  will  result 


V .  .  cos  a,  =  v^ 


(197) 


whence  the  angular  velocity  about  any  axis  oblique  to  the  instanta- 
neous axis,  is  equal  to  the  angular  velocity  of  the  body  multiplied 
by  the  cosine  of  the  inclination  of  the  two  axes. 

|X  §162. — Equation  (196)  gives  v.,  when  v^,v^,v, ,  are  known.  To 
find  these,  resume  Equations  (US),  and  write  for  the  moments  of  the 
extraneous  forces  in  reference  to  the  axes  a;,'  y,'  2,'  through  the  centre 
of  inertia,  iV^,  3/^,  Z^,  respectively,  then  will 


C-.... 


(/  t 


^■^) 


2  m 


A' 


\  dt"  d  I"        J 

■V  dt^      -^  d  P         /  ' 


(198) 


differentiating  the  first  of  Equations  (192),  with  respect    to  t,  .we  find 


d-^x'  _         d£_ 

ITfi  ~  ^y'  dt 


dt  dt 


dt      -^ 


d  z^  d  ?/' 

and  replacing  — ; —  and  —, —  i  by    their    values    given    in    the    second 
'^         ^    dt  dt        ^  *= 

and  third  of  Equations    (192), 

d-^x' 


/72  yl  /  \ 


dt  dt     ^ 


MECHANICS     OF     SOLIDS. 


155 


in  the  same  way 


dr~  z' 


and  these  values  in  the  first  of  Equations,  (198),  give 


d  t 


+ 


Im.z'x' 


!.  =  X,.(190) 


\y  '   dtJ 


Similar  equations  will  result  from  the  remaining  two  of  Equations 
(198);  then  by  elimination  and  integration,  we  might  proceed  to  find 
the    s^alues    of  v^,    v     and    v^ ,    but    the    process    would   be   long   and 

tedious.  It  will  be  greatly  simplified,  however,  if  the  co-ordinate 
axes  be  so  chosen  as  to  make  at  the  instant  corresponding  to  t, 


-Emx'  i/  =  0;    2  m  z'  y'  =  0  ;    2  m  z'  x 


0 


(200) 


which    is    always    possible,    as    will    be    shown    presently.      This  will 
reduce  Equation  (199)  to 


di 


(y'.  +  .^'2)  +  v^  .  v^  .  2  m  (.^■'2  -  y'2)  z=  L, 


The    other    two     equations   which    refer   to   the   motion    about   the 
axes  y'  and  x\  may  be  written  from    this   one.     They  are, 

^y  .  2  m  (.i-'2  +  2'2)  _^  v^  .  V,  .  2  m  (2'2  -  a;'^)  =  M, , 

^^  .  2  7?i  (y'2  +  g'2)  +  v^  .  v^  .  2  m  (v'2  --  £'2)  =  iV^ . 


156  ELEMENTS     OF     ANALYTICAL     MECHANICS. 

The  axes  x',  y',  z\  which  satisfy  the  conditious  expressed  in 
Equations  (200),  are  called  the  'priacrpal  axes  of  Jigure  of  the  body. 
And  if  we   make 


2  m  .  (y'2  4-  .r'2)  ^  Q, 
2  m  .  {x'-  +  s'2)  —  B, 
2  m  .  (y'2  +   ^'2)  ^  ^i  . 

we  find,  by  subtracting    the    third    from    the   second, 
2  m  .  (.t'2  -  T/'s)  -  B  ~  A, 

the    first   from    the    third, 

2  m  .  (^'2  _  ;^'2-)  ^  ^  _  (7^ 

and    the    second   from   the    first, 

2m.(?/'2  _  .'2)  =   C-  B; 


which    substituted    above,   give. 


dv 


(201) 


^•^;  +  v„.v,.((7-i?) 


JV. 


(202) 


By  means  of  these  equations,  the  angular  velocities  v^ ,  v  ,  v^ ,  must 
be  found  by  the    operations  of  elimination    and  integration, 

§  103. — It  is  plain  that  the  Cjuantities  C,  B  and  A,  are  constant 
for  the  same  body  ;  the  first  being  the  sum  of  the  products  arising 
from  multiplying  each  elementary  mass  into  the  square  of  its  dis- 
tance from  the  principal  axis  z\  the  second  the  same  for  the  prin- 
cipal axis  y',  and  the  third  for  the  pi'incipal  axis  x'.  The  sum 
of  the  products  of  the  elementary  masses  into  the  square  of  their 
distances  from    any  axis,  is   called   the  moment  of  inertia  of  the  body 


MECHANICS     OF     SOLIDS 


157 


in  reference  to   this  axis.     A,  B  and   C  are  called  principal  momenta 
of  inertia. 

§  164. — TliiouLih  any  assumed  point  tlierc  may  always  be  drawn 
one  ,set  of  rectangular  axes,  and,  in  general,  only  one  whieli  will  satisfy 
tlie  conditions  of  Equations  ('200).  To  show  this,  assume  the  formulas 
for  tlie  transformation  from  one  system  of  rectangular  axes  to  another, 
also  rcctan^'ular.     These  are 


x'  =  .T.cos  [x'  x)  +  y  cos  {x'  y)  +  z  cos  [x'  z),^ 
y'  =  x  cos  {y' x)  +  y.cos  {y' y)  +  s.cos  {y' z), 
z'  —  X  cos  {z'  x)  +  y  cos  {z'  y)  -\-  z .  cos  [z'  2), 


(203) 


in  which  (x'  x),  {y' •^)  ^^^'^  (^''^)'  denote  the  angles  which  the  new 
axes  x',  y',  z\  make  with  the  primitive  axis  of  x\  {x' y),  {y' y) 
and  (s' y),  the  angles  which  the  same  axes  make  with  the  primitive 
axis  of  y,  and  {x'  z),  {y'  z)  and  {z'  z),  the  angles  they  make  Avith  the 
axis  z. 

Assume  the  common 
origin  as  the  centre  of  a 
sphere  of  which  the  radius 
is  unity  ;  and  conceive  the 
points  in  which  the  two 
sets  of  axes  pierce  its  sur- 
foce  to  be  joined  by  the 
arcs  of  great  circles  ;  also 
let  these  points  be  con- 
nected with  the  point  iV, 
in    which    the    intersection 

of    the    planes   xy   and   x' y'   pierces    the  spherical  surflice  nearest  to 
that  in  which  th-e  positive  axis  x  pierces  the    same.     Also,  let 
6  :=  Z'  AZ  =z  X'  NX,  being  the  inclination  of  the  plane  x'  y'  to  that 

of  X  y. 
•\,  =  NAX  being    the    angular   distance   of    the    intersection    of    the 

planes  x  y  and   x'  y\  from  the    axis   x. 
(p  —  N AX'  being    the     angular    distance    of   the    same    intersection 

from   the    axis   x' . 


158  ELEMENTS     OF     ANALYTICAL     MECHANICS 

Then,  in    the    spherical    triangle  X'  NX^ 

cos  {x'  x)  =  cos  -^j^  •  cos  (p  -f  sin  Y  .  sin  9 .  cos  &  j 

In  the  triangle  Y'  NX,  the  side  N  Y'  =  ^  +9,  and 

cos  (y'  .r)  =  —  cos  4^ .  sin  9  -^  sin  ^^ ,  cos  9  .  cos  &  , 

In   the   triangle  Z'  NX,  the  side  NZ'  =  -^,   and 

cos  [z'  x)  =  sin  %[/ .  sin  6. 

And   in   the   same   way  it  will   be  found  that 

cos  [x'  y)  =  —  sin  -^ .  cos  9  +  cos  -^  .  sin  9 .  coi  i  5 
cos  [y'  y)  =  sin  ■>]> .  sin  9  +  cos  ■\/ .  cos  9  .  cos  ^  ; 
cos  {z'  y)  =  cos  -s}^ .  sin  &  ; 
cos  [x'  s)  =  —  sin  9  .  sin  6  ; 
cos  (?/'  0)  =  —  cos  9  .  sin  ^  ; 
cos  [z'  z)  =  cos  ^  ; 

and    by   substitution    in    Equations  (203), 

x'  =:  X  (sin  4^ .  sin  9  .  cos  6  +  cos  4^ .  cos  9) 

+  y  (cos  -4/ ,  sin  9  .  cos  6  —  sin  4^  •  cos  9)  —  z  sin  9  .  sin  d, 
y'  =  X  (sin  4^  •  cos  9  .  cos  ^  —  cos  4-  •  si»  9) 

+  y  (cos  4^  .  cos  9  .  cos  ^  +  sin  4^  ■  sin  9)  —  2  cos  9  .  sin  ^, 
z'  =:  X  sin  4^  •  sin  &  -\-  y  cos  4^ .  sin  ^  +  2  cos  &  ; 

or    making,  for  sake  of  abbreviation, 

D  =:  X  cos  4^  —  y  sin  4^, 

E  —  X  sin  4> .  cos  &  -{-  y  cos  4^ .  cos  ^  —  2  sin  ^, 

the  above    reduce   to 

x'  ~  E .  sin  9  +  J9  .  cos  9, 
y'  z=  E .  cos  (p  —  D  .  sin  9, 
z'  =  X  .  sin  4^ .  sin  6  -{-  y  .  cos  4^ .  sin  ^  +  z .  cos  ^. 


MECHANICS     OF    SOLIDS.  159 

Substituting   these  values  in   the   equations 

2  m  .  .r' .  y'  =  0  ;     ^m.z'  .s'  =  0;     2  m  .  y' .  2'  =i  0  ; 
we  obtain  from  the  first, 

sin  cp  .  cos  £p .  2  ??i  (^2  _  2J2)  -{-  (cos- <p  —  sin^  up)  Z7nE.D  =  0, 

or,  replacing  sin  9.  cos  9,  and  cos^^  —  sin2(p,  by  their  equals  -J  sin  2  (p, 
and  cos  2  9,  respectively, 

sin2ip.2m(^2  _  x)2)  +  2  cos  2(p  .  2  mZ> .  ^  =  0;  •  •  •  (204) 

and  from    the    third    and   second,  respectively, 

cos  9  .  2  ?7i .  ^ .  s'  —  sin  9  .  2  m  D  .  s'  =  0,  •     •     •     (205) 
sin9  .2m.^.  s'+  cos9.2mi).2'  =  0.  •     •     •     (200) 

Squaring  the  last  two  and  adding,  we  find 

(2  m  .  E.  z'f  +  (2  m  .  Z> .  z'f  =  0. 
which  can  only  be  satisfied  by  making 

Im.U.z'  =  0:) 

'[ (20V) 

These  equations  are  independent  of  the  angle  9,  and  will  give  the 
values  of  4'  and  6 ;  and  these  being  known.  Equation  (204)  will  give 
the  angle  9. 

Eeplacing  U  and  D  by  their  values,  we   have 

E  .z'  —  sin  ^  .  cos  ^  [xr  sin-  -j/  +  2  .r  y  sin  -j'  cos  -^i  ■\-  y"^  cos^  ^z  —  z"^) 
+  (cos^  ^  —  sin^  &)  {x  z  sin  -^j  -\-  y  z  .  cos  4^)  , 

D  .z'  =1  sin  ^  \xy  (eos^'  ■\>  —  sin^  4^)  +  {x^  —  y'^)  sin  4^  cos  4'} 
4"  cos  S  (.T  2  cos  4^  —  yz  sin  4^)  • 

and  assuming 

2  m  .r2   =  A'-,  2  in  ^f  —  B' ;  I,  m  z-  =   C ; 
1.mxy  =  E';  I^mxz  —  F' ;  'S.myz  —  H', 

and  replacing  sin  &  .  cos  ^,  and  cos-  ^  —  sui^  ^^  by  their  respective 
values,  ^  sin  2  ^,  and  cos  2  (?,  Equations  (207)  become 

sin  2  &  {A'  sin2  4.  +  2  ^'  sin  4.  cos  4.  +  B'  cos^  s}.  -  C) 


+  2  cos  2  ^  (i?''  sin  4.  4-  H'  cos  -4)  '  ' 


160  ELEMENTS     OF    ANALYTICAL    MECHANICS. 


sia  &{E' .  (cos2.].  —  sin2  ^)  +  {A'  —  B') .  sin  4.  cos  -^ 
-f-  cos  (5  (i^'  cos  ■].  —  H'  sin  4) 


'[.., 


in  which  ^',  B',  C,  E\  F'  and  H\  are  constants,  depending  only 
upon  the  shape  of  the  body  and  the  position  of  the  assumed  axes 
X,  y,  z. 

Dividing    the    first    by    cos  2  ^,  and    the    second    by    cos  ^,    they 
become 

tan  2  ^  .  (^'  sin2  J.  +  2  ^'  sin  -1  cos  X  +  B'  cos  2  J.  —  C'l  )  ^     ^^ 

+  2  {F  sin  4.  +  ^'  cos  vl)  ) 

tan  ^  .  1^'  (cos2  X  —  sin2  -1)  +  (^'  -  B'\  sin  -1  cos  ^L}■  ) 

\       \         -r  r)    r  y  ;         -^         Yi  (  ^0.(207)" 

.  +  i^'  cos  4.  -  H'  sin  4.  i 

From  the  first  of  these  we  may  find  tan  2  ^,  and  from  the  second, 
tan^,  in  terms  of  sin%|>,  and  cos  4/-,  and  these  values  in  the  equation 

2  tan  ^ 
tan  2  ^  =  ^-—^, (208) 

will  give  an  equation  from  which  4^  ™-iy  l^e  found. 
In  order  to  effect  this  elimination  more  easily,  make 

tan  4^  =  ■?', 
whence 

u  "* 

sin  4^  =  — -p==  \  cos  4^  = 


making  these  substitutions  above,  we  find 


A'  ifi  +  2  ii"  ?<  4-  B'  —  6"  (1  +  m2) 


tan  ^  =  ^ 


E'  (1  -  zt2)  _(_  (^4'  _  i^')  ^^ 

which  in  Equation  (208)  give 

(         B'F'-F'C'-E'H'     )   1 
^E'il-^,)HA'-B')u)  I  ^  (^,,,,_^,,,,,+^,^,)„  M  ^  0 


(209) 


MECHANICS     OF     SOLIDS.  161 

which  is  an  equation  of  the  third  degree,  and  must  have  at  least 
one  real  root,  and,  therefore,  give  one  real  value  for  -j/.  This  value 
being  substituted  in  either  of  the  preceding  equations,  must  give  a 
real  value  for  6.  and  this  with  -v]^,  in  either  of  the  Equations  (205) 
or  (206),  a  real  value  for  cp  ;  whence  we  conclude,  that  it  is  always 
possible  to  assume  the  axes  so  as  to  satisfy  the  required  conditions, 
and  that  through  every  point  there  may  be  drawn  at  least  one  set  of 
principal  axes  at  right  angles  to  each   other. 

The  three  roots  of  this  cubic  equation  are  necessarily  real ;  and 
they  represent  the  tangents  of  the  anglgs  which  the  axis  x  makes 
with  the  lines  in  which  the  three  co-ordinate  planes  x'  y',  y'  z\  x'  z\  cut 
that  of  a;?/;  for  there  is  no  reason  why  we  should  consider  one 
of  these  angles  as  given  by  the  equation  rather  than  the  others,  and 
the  equations  of  condition  are  satisfied  when  we  interchange  the 
axes  x'  y'  z'.  Hence,  in  general,  there  exists  only  one  set  of  prin- 
cipal axes.  If  there  were  more,  the  degree  of  the  equation  would 
be  higher,  and  would,  from  what  we  have  just  said,  give  three  times 
as    many  real   roots  as  there  are  systems. 

If  E'  =  H'  —  F'  —  0,  Equation  (209)  will  become  identical ;  the 
jiroblem  will  be  indeterminate,  have  an  infinite  number  of  solutions, 
and  the  body  consequently  an  infinite  number  of  sets  of  principal 
axes.     Such  is  obviously  the   case  with    the   sphere,  spheroid,  &c. 

MOiMENT    OF   rS^EKTIA,    CENTRE   AND   RADIUS   OF    GYRATION. 

§  165. — The  quantities  A.  B  and  (7,  in  Equations  (201)  are  the 
moments  of  inertia  of  the  body  in  reference  to  the  principal  axes. 
To  find  these  moments  in  reference  to  any  other  axes  having  the 
same  origin  as  the  principal  axes,  denote  by 

x\  y\  z'^  the  co-ordinates  of  m  referred    to    the   principal  axes ;  by 
.r,  y,  £■,  the  -co-ordinates    of    the    same    element   referred    to    any 
other   Rectangular  system  having  the  same  origin  ;  and  by 
C",  the   moment  of  inertia  referred  to  the   axis  z  ; 
then  from  the  definition, 

C"  =  2  ni .  (.1-2  +  ?/2)  —  2  m  a."  +  2  /«  ?/2  . 
11 


1G2  ELEMENTS     OF    AXALYTICAL    MECHANICS, 

but   by    the   usual  formulas  for   transformation, 

X  =1  a  x'  -\-  h  y'  -{-  c  z', 
y  =r  a'  x'  +  h'y'  -\-  c'  z', 
z  =  a"x'.+  h"y'  +  c"z\ 

in  which  a,  6,  &c.,  denote  the  cosines  of  the  angles  which  the  axes  of 
the  same  name  as  the  co-ordinates  into  which  they  are  respectively 
multiplied  make  with  the  axis  corresponding  to  the  variable  in  the 
first  member. 

Substituting  the  values  of  x  and  y  in  that  of  C",  and  reducing  by 
the   relations, 

2  m  .t'  ?/'  =  0  ;     2  m  x' z'  =  0  ;     2  m  y'  s'  =  0  ; 
and  we   have, 

C"  =  a"^.:Em{y'^  +  z'^)  +  i"^  .  2  m  (.x-'2  +  ^'2)  +  c"K  2  m  {x'^  +  y''') ; 
and  by  substituting  A,  B  and   C  for  their  values,  this  reduces  to 

C  =  a"'^A  +  h"^B  +  c"2  C     '     '     •     •      (210) 

That  is  to  say,  the  moment  of  inertia  with  reference  to  any  axis 
passing  through  the  common  point  of  intersection  of  the  principal 
axes,  is  equal  to  the  sum  of  the  products  obtained  by  multiplying 
the  moment  of  inertia  with  reference  to  each  of  the  principal  axes, 
by  the  square  of  the  cosine  of  the  angle  which  the  axis  in  question 
makes  with  these  axes. 

8  166. — Let  A,  be  the  greatest,  and  (7,  the  least  of  the  moments 
of  inertia,  with  reference  to  the  principal  axes ;  then,  substituting  for 
a"'\  its  value,  1  —  V^  —  c"2,  in  Equation  (210),  we  have 

C  =  A  -  b"^A  -  B)  -  c"^A  -  C).      '     •      (211) 

By  hypothesis,  A  —  B,  and  A  —  C,  arc  positive-,  therefore,  C  is 
always  less  than  A,  whatever  be  the  value  of  b",  and  c". 

Again,  substituting  for  c"^  its  value  1  -a"^  *-  b""-  in  Equation 
(210),  we  get 

(7'  =  (7  +  a"2  (.4  -  C)  +  b"~  {B-  C)      •     •     •     (212) 
and  C"  must  always  be  greater  than   C. 


MECHANICS     OF    SOLIDS.  1G3 

Whence,  we  conclude  that  the  principal  axes  give  the  greatest  and 
least  moments  of  inertia  in  reference  to  axes  through  the  same  point. 
If -d  be  equal  to  B,  then  will  Equation  (211)    become 

(7'  =  (1  -  c"-')  A  +  c"2  C, (2_133 

and  this  only  depending  upon  c",  we  conclude  that  the  moment  of 
inertia  will  be  the  same  for  all  axes  making  equal  angles  with  the 
principal  axis,  z'.  The  moments  of  inertia,  with  reference  to  all  axes 
in  the  plane  x'  y\  are,  therefore,  equal  to  one  another.  But  all  the 
axes  in  the  plane  x'  y\  which  are  at  right  angles  to  one  another, 
are,  §  164,  when  taken  with  z\  principal  axes,  and  we,  therefore, 
conclude  that  the  body  has  an  indefinite  number  of  sets  of  principal 
axes. 

If,   at   the   same   time,  we   have  A  ^^  B  ^  C,  then  will    Equation 
(210)  reduce  to 

C  ^  C  =  A  =  B. 

that  is,  the  moments  of  inertia  are  all  equal  to  one  another,  and  iA\ 
axes  are  principal,  the  Equation  (210)  being  satisfied  independently 
of  a",  6",  c". 

§  16*7. — Resuming    Equations,     (33),    and    substituting    the    values 
of  a*,  y,  2,  in  the  general  expression, 

which  is  the  moment  of  inertia  with  reference  to  any  axis,  2,  parallel 
to  the   axis  2',  through  the  centre  of  inertia,  we  hav6 

2  m  {x^  +  y-^)  z=  2  m  [  (:c,  +  x'f  +  {v ,  +  y'Y\ 

=  2  m  {x''^  +  ?/'2)  +  {x^  +  yf) .  2  m 
+  2  a;,  .  2  ?/i x'  +  2  y,  .  2  viy' ; 

but  from  the   principle  of  the  centre  of  inertia, 

2  ma;'  =  0,     and     "Lmy'  —  ^\ 

whence,  denoting  by  d  the  distance  between  the  axes  z  and  s',  and 
by  M  the  whole  mass, 

2  m  .  (a;3  +  y^)  =  H  m  {x'^  +  y'2)  _|_  ji/c^2  .     .     .     (.214) 


lOi  ELEMENTS     OF     ANALYTICAL    MECHANICS. 

That  is,  the  moment  of  inertui  of  any  body  in  reference  to  a  given 
axis,  is  equal  to  the  moment  of  inertia  ^vith  reference  to  a  parallel 
axis  through  the  centre  of  inertia,  increased  by  the  product  of  the 
whole  mass  into  the  square  of  the  distance  of  the  given  axis  from 
that   centre. 

And  we  conclude  that  the  least  of  all  the  moments  of  inertia  is 
that  taken  with  reference  to  a  principal  axis  through  the  centre  of 
inertia. 

§  ICS. — Denote  by  r  the  distance  of  the  elementary  mass  m  from 
the  axis  2,  then  will 

r2   r=   a;2   _[-   2/2j 

and 

Now,  denoting  the  whole  mass   by  J/,  and  assuming 
2  m  r2  1=  Mk\ 


we   have 

k 


/2  iiir^  i„-,.. 


The  distance  Tc  is  called  the  radius  of  gyration^  and  it  obviously 
measures  the  distance  from  the  axis  to  that  point  into  which  if  the 
whole  mass  were  concentrated  the  mouicnt  of  inertia  would  not  be 
altered.  The  point  into  which  this  concentration  might  take  place 
and  satisfy  the  condition  above,  is  called  the  ceiitre  of  gyration. 
When  the  axis  passes  through  the  centre  of  inertia,  the  radius  k 
Olid  the-  point  of  concentration  are  called  p^-inc'qoal  rudius  and  ^:);'i«- 
cipal  centre  of  gyration. 

The  least  radius  of  gyration  is.  Equation  (215),  that  relating  to 
the  principal  axis  with  reference  to  which  the  moment  of  inertia  is 
the  least. 

If  k^  denote  a  principal  radius  of  gyration,  we  may  rephice 
2??c.  (.i;'2  -J-  ?/'2)  in  Equation  (214)  by  Mk^"^^  and  we  shall  have 

2wr2  =3/P  =  iJ/(/.-,2  _f.  c;-.)      .     .     .     .     (21C) 


MECHANICS     OF     SOLIDS. 


165- 


If  the  linear  dimensions  of  the  body  be  very  small  as  compared 
with  d,  we   may   write  the   moment  of  inertia  equal    to  M(P. 

The  letter  k  with  the  subscript  accent,  will  denote  a  principal 
radius  of  gyration, 

§169. — The  determination  of  the  moments  of  inertia  and  radii 
of  gyration  of  geometrical  figures,  is  purely  an  operation  of  the  cal- 
culus. Such  bodies  are  supposed  to  be  continuous  throughout,  and 
of  uniform  density.  Hence,  we  may  write  dM  for  m,  and  the  sign 
of  integration  for  2,  and    the  formula  becomes 


2  in  r^ 


-  /^^^l^- 


(217) 


Example   1. — A  physical  line  about  an   axis    through   its   centre  and 
perpendicular    to    its    length. 


Denote    the  whole   length   by  2a;    then 
2a  :dr:\  M :  d  M, 


whence. 


and 


dr 
dM  =z  M-  --, 

2a 


Mk 


.-/' 


^M-~-dr 

2  a 


Ma^ 


If  the    axis  be   at  a   distance  d    from    the   centre,  and    parallel    to 
that    above,  then,  Equation  (21G), 

h  =  ^/\^K^\^~d?. 

Example  2. — A    circular   plate    of   uniform   density   and    thickness, 
about    an  axis    tliruugh    its   centre   and  perpendicular    to  its   plane. 


1G8  ELEMENTS     OF     ANALYTICAL     MECHANICS. 

Denote  the  radius  by  a ;  the  angle  X A  Q 
by  ^  ;  the  distance  of  d  M  from  the  centre 
by  r ;    then, 


ifa'^:r.d6.dr  :  :  M :  d M; 


whence. 


dM  =  M- 


r  .d  r .  d 6 


and 


Mk/  =  3f--^^-dd 


=   /    2  il/.  —  •  c?  ?•  =  — --, 
t/  0  a^  i2 


and  for  an  axis   parallel  to   the  above  at  the  distance  cZ, 


h  =  -v/|a2  +  d^ 

Example  3. — The  same  body  about  an  axis  through   its   centre  and 
in    its  plane. 


As  before, 


dM  =  M- 


r  .dr . d d 


m.  ^\  hich  r  denotes  the  distance  of  d  31  from  the  centre  ;  and  taking 
the  axis  to  be  that  from  which  6  is  estimated,  the  distance  of  ciie 
elementary  mass  froin    the   axis  will    be   r  sin  6,  and 


Mk. 


and 


2=/     /     M-^-~--dr.d&^-^J     /     r3(]  -cos2(5)Jr.f/c), 


3Ik^  =  ^  /    rKdr  =M-. 
'         a^  Jo  4 


k,  =  ^0, 
and  about   an    axis   parallel  to  the    above  and    at   the   distance  d. 


V   4 


—  a2  -j-  (^2, 


MECHANICS     OF     SOLIDS. 


167 


It  is  obvious  that  both  the  axes  first  considered  in  Examples  2 
and  3  are  principal  axes,  as  are  also  all  others  in  the  plane  of 
the  plate  and  through  the  centre,  and  if  it  were  required  to  find 
the  moment  of  inertia  of  the  plate  about  an  axis  through  the  centre 
and  inclined  to  its  surflice  under  an  angle  9,  the  answer  -would  be 
given  by  the  Equation  (210), 

Mk;^  =  \  Ma-  sin2  9  +  i  Mii?  cos2  9 
=  iJ/a2(l  +  sin2(p), 

and  for  a  parallel  axis  whose    distance  is  0?, 

Mlfi  =  IT  (i    a3  (1  +  sin2(p)  +  d?)  . 

Example  4. — A  solid  of  revolution  about  any  axis  2^S'''P^^^dicular  to 
the    axis    of  tlie    solid. 

Let  J)  A'  E  be  the  given  axis, 
cutting  that  of  the  solid  in  A'-  Let 
A'  be  the  origin  of  co-ordinates, 
F  M  ^  y;  A'  F  =  x  ;  A  A'  =  m  ■ 
A'  B  =  n  ;  and  V  =  volume  of  the 
solid. 

The    volume     of    the     eleinentary 
.  section  at  F  will  be 


and 

whence. 


If  y"^  .  d  X, 

V  :  M  ::rr  .y"^  .dx:  dM; 

<^''  M  =  —  "X  '  y~  •  d  ,r, 


and  its  moment  of  inertia  about  M  M\  is,  Example  o, 

and  about  the  parallel  axis,  D  E, 
M 


16S  ELEMENTS     OF    ANALYTICAL    MECHANICS, 

therefore. 


But 


whence, 


MF^  =  Jl  Y^'-y'  (4  2/-  +  •^■')  ^  ^' 


k^  = 


£y''' 


The   equation    of  the    generating  curve   being    given,  y  may  be  elimi- 
nated and  the  integration  j^erformed. 

Exam'ple   5. — A  sphere   about   a   line    tangent   to  its   surface. 
The  equation  of  the  generatrix  is 

y^  =  2  a  a;  —  a;^  ; 

m  -which  a  is  the  radius  of  the  sphere.     Substituting    the  value  of  y"^ 
in  the  last  equation,  recollecting  that   m  =  0,   and  n  —  2  a,  we  have 

r  "(a2  .^2  4-  a  .1-3  -  f  x^)  d  X 

y-2a  5 

(2  a  X  —  x^)  d  x 


Also  Equation  (210), 


A;  2  =  A-2  _  a2  =  I  a2 
5 


and 

.^1  =  «' 

Thus,  when  tlie  boundary  of  a  rotating  body  and  the  law  of  its 
density  may  be  defined  by  equations,  its  moment  of  inertia  is  readily 
found  by  the  ordinary  operations  of  the  calculus ;  but  when  tlie  iigui'e 
is  irregular  and  the  density  discontinuous,  I'ecourse  is  had  to  the  prop- 
erties of  the  compound  pendulum,  to  be  explained  presently. 


MECHANICS     OF    SOLIDS.  109 

JExamide  6. — Find  the  j-'oints  i)i  reference  to  which  ihe  iirincipdl  mo- 
ments are  equal. 

Take  the  origin  at  the  centre  of  inertia,  and  the  principal  axes 
through  that  point  as  the  co-ordinate  axes.  Denote  by  x^  y ^  z^  the  co- 
ordinates of  one  of  the  points  sought ;  hy  -1^,  -Z>^,  and  C^  the  principal 
moments  with  reference  to  this  point,  and  by  x'  y'  z'  the  co-ordinates  of 
the  element  m.  Then,  because  the  moments  through  the  point  x  v  z 
are  to  be  principal,  .ill  i^C^^'^'-'^'V- '^/V  •-^^'  V-j 

^m{x'-x^{j/'~y)  =  Q-,^m{x'-x;){z'-z;)  =  0\^m{y'-y;){z'-z^-=.0. 

Performing  tlie  multiplication  and  reducing  by  the  properties  of  the 
centre  of  inertia  and  principal  axes,  we  have 

M.  x^  y^  =  0  ;   Mx^  2,  =  0  ;   My^  z^  =  0: 

which  can  only  be  satisfied  by  making  two  of  the  co-ordinates  x^y^z^ 
separately  zero.     Let  y^  =  0.  and  z^  =  0  ;    then,  §  166  and  E(j.  (21(i), 

A^^A;    B^  =  B  +  i/.r/  ;    C,  =  C  +  Mx^' ; 

but,  by  the  conditions,  the  first  members  are  equal.     Whence 

A^B  +  Mx;  =  C+  Mx;  ; 
and,  therefore, 

B=  G-    and  x^  =  ±\/^^^^- 

and  from  which  it  is  apparent  :  Ist,  that  if  all  the  principal  ujomcnts 
in  reference  to  the  centre  of  inertia  be  uncijual,  there  is  no  point  in 
reference  to  which  they  can  be  equal  ;  2(1,  that  if  two  of  them  bo 
equal  in  reference  to  the  centre  of  inertia  and  the  third  be  the  great- 
est, there  are  two  points,  equally  distant  from  the  centre  of  inertia  and 
on  the  axis  of  the  greatest  momeu.t,  with  reference  to  which  tliey  are 
equal ;  od,  that  if  all  three,  with  reference  to  the  centre  of  inertia,  be 
equal  to  one  another,  there  is  no  other  point  with  respect  to  which 
they  can  be  equal. 

IMPULSIVE    FORCES. 

§  1"«^ — We  have  thus  far  onlv  been  concerned  with  forces  whose 
action  may  be  likened  to,  and  ind-i'd  repi'csented  by,  the  pressure 
arising    from    the    weight    of    some    definite    bodv,   as    a    cubic    lout    of 


170  ELEMENTS     OF    ANALYTICAL    MECHANICS. 

distilled  water  at  a  standard  temperature.  Such  forces  are  called 
incessant,  because  they  extend  their  action  through  a  definite  and 
measurable  portion  of  time.  A  single  and  instantaneous  effort  of 
such  a  force,  called  its  intensity,  is  assumed  to  be  measured  by  the 
whole  effect  which  its  incessant  repetition  for  a  unit  of  time  can 
produce  upon  a  given  body.  The  effect  here  referred  to  is  called 
the  quantity  of  motion,  being  the  product  of  the  mass  into  the 
velov^ity  generated.     Tliat  is,  Equations  (12)  and  (13), 

P  =  3I.K=M'^=Mg^;    ....   (2KS) 

in  which    F^,  denotes  the  velocity  generated  in  a  unit    of  time. 

The  force  F,  acting  for  one,  two,  or  more  units  of  time,  or  for 
any  fractional  portion  of  a  unit  of  time,  may  communicate  any  other 
velocity  V,  and  a  quantity  of  motion  measured  by  M  V.  And  if 
the  body  which  has  thus  received  its  motion  gradually,  impinge  upon 
another  which  is  free  to  move,  experience  tells  us  that  it  may 
suddenly  transfer  the  whole  of  its  motion  to  the  latter  by  what 
seems  to  be  a  single  blow,  and  although  we  know  that  this  transfer 
can  only  take  place  by  a  series  of  successive  actions  and  reactions 
between  the  molecular  springs  of  the  bodies,  so  to  speak,  and  the 
mertia  of  their  different  elements,  yet  the  wliole  effect  is  produced  in 
a  time  so  short  as  to  elude  the  seises,  and  we  are,  therefore,  apt  to 
assume  though  erroneously,  that  the  effect  is  instantaneous.  Such 
an  assumption  implies  that  a  definite  velocity  can  be  generated  in  an 
indefinitely  short  time,  and  that  the  measure  of  the  force's  intensity 
is,  Equation  (21S),  infinite. 

In  all  such  cases,  to  avoid  this  difficulty,  it  is  agreed  to  take  the 
actual  motion  generated  by  these  blows  during  the  entire  period 
of  their  action,  as  the  measure  of  thuir  intensity.  Thus,  denoting 
the  mass  impinged  upon  by  i/,  and  the  actual  velocity  generated 
in    it   when   perfectly   free   by    V,   we   have 

F=^  MV    =  M.p^, (219) 

in    which   P,    denotes  the    intensity   of    the     force's    action,    and    the 
second  member  of  the  equation  the  resistances  of  the  body's  inertia. 


MECHANICS     OF    SOLIDS. 


171 


Forces  which   act   in  the   manner   just   described,    by    a   blow,    are 
called   impulsive  forces. 

MOTION    OF   A   BODY   UXDEE   THE   ACTION   OF   IMPULSIVE   FORCES. 

§  1*70. — The  components  of  the  inertia  in  the  direction  of  the  axes 
X  y  z,  are  respectively 

,^  ds  dx        ,,   dx 
M-- — —  —  M'--; 
dtds  d  t 

a  t    a  s  d  t 

^^  ds   dz        -.r  dz 

M'-, —  =  M'  —  ; 

dtds  d  t 

which,    substituted   for    the    corresponding    components    of   inertia   in 
Equations  (^1)  and  (i?),  give 

7-.  dx    ^ 

2  P  cos  a  =  2  m  •  -;-  *, 
dt 


/-.  dy     , 

2  P  cos  ^  =  2  »i  .  -^ ;    > 
dt 


2  P  cos  7  =  2  ?ii  •  —  ; 
d  i     ^ 


(220) 


(221) 


2  P  {x'  cos  /3  —  y'  cos  a)  ==  2  m  (x'  •  ^  —  y'  •  ^)  , 
2  P  U'  cos  a  —  x'  cos  r)  =:  2  m  (  2'  •  ^  —  x'  ■  ~) 

^  ' '  \      dt  dty 

2P(/cos7  -sr'cos^)  =  2?«  {y'  .'-±— z'  .'^  . 

In  which  it  will  be  recollected  that  x  y  z  are  the  co-ordinates  of  m^ 
referred  to  the  fixed  origin,  and  x'  y'  z',  those  of  the  same  mass 
referred  to  the  centre  of  inertia. 


MOTION   OF   THE   CENTRE     OF    INERTIA. 

§171. — Substituting    in    Equations    (220),    for    dx,    dy,  d  z,    their 

values   obtained    from  Equations  (34),  and  reducing  by  the  relations 

^mdx'  =  0;    ^mdy'  =  0;   :^indz'  =  0;    -  •     .   (222) 


172         ELEMENTS     OF     ANALYTICAL    MECHANICS 
given  by  the  principle  of  the  centre  of  inertia,  we  find 


2  P  cos  a 


dx, 
d  t  '  ^ 


2  P  COS  /3  =  -^  ■  2  ?/j :    ). 

2  P  cos  r  rr  -— ^  •  2  ??i : 
'  d  t 

and  substituting  M  for  2  in,  we   have 


2  P  cos  a  =  i/ 


2  P  cos  /3  =  if 


2  P  cos  7  =  J/  .  ^ ; 


(223) 


which  are  wholly  independent  of  the  relative  positions  of  the  elements 
of  the  body,  and  from  which  we  conclude  that  thr,  motion  of  the 
centre  of  inertia  will  be  the  same  as  though  the  mass  were  concen- 
trated in    it,   and  the    forces    applied  immediately  to   that  point. 

§  1*72. — Eeplacing  the  first  members  of  the  al)Ove  equations  by 
their  values  given  in  Equations  (41),  and  denoting  by  V  the  velocity 
which  the  resultant  R    can  impress  upon  the  whole  mass,  then  will 

2  y-*  cos  a  =  J/ F  cos  a;    2P  cos/3=  if  Fcos  i  ;    2  P  cos/ =  i/ Fcos; : 


Buustituting   these    above,  we  find 

F. cos  a  = 
F . cos  h  ■= 
V .  cos  c  = 


d  x^ 

~dJ  ' 

dt    ' 

dz^_ 
dt    ' 


(224) 


MECHANICS     OF    SOLIDS. 


173 


and  integrating, 


x^  =  V-cosa.t  +  C,    ' 

tj^  =  V.  cos  b.i  +   C",     I (225) 

z,  =:  V.cosc.t  +  C",  ^ 

and   eliminating    t   from    these    equations,    V  ■will   also  disappear,    and 
we  find, 

rc05-CA        C"  cos  c  —  C"  cos  a     " 

z.  =  x.-\- ) 

Icpsal 


2/  =  y/ 

yy    =   •'«^; 


cos  a 
COS  c        C"  cos  c  —  C"  cos  6 


COS  b 

cos  i 
COS  a 


cos  6 

C  cos  6  —  C"  cos  a 


,     I 


(22G) 


which  being  of  the  first  degree  and  cither  one  but  the  consequence 
of  the  other  two,  are  the  equations  of  a  straight  line.  This  line 
makes  with  the  axes  x,  y,  z,  the  angles  a^^,  3  respectively,  and  is, 
therefore,  j)arallel    to    the   resultant  of  the  impressed  forces. 

Whence  we  conclude,  that  the  centre  of  inertia  of  a  body  acted 
upon  simultaneously  by  any  number  of  impulsive  forces,  will  move 
uniformly  in  a  straight  line  parallel  to    their    common   resultant. 


MOTION    ABOUT   THE   CENTRE   OF   INERTIA. 

§  173. — Substituting,  in  Equations  (221),  for  dx,  drj  and  dz^  their 
values    from    Equations    (34),    reducing   by 

2  m  x'  =  0, 

2  m  y'  —  0, 

2  m  z'  —  0, 
and  we  find, 

T.  ,  ,        n  ,  X  /'  ,     dy'  ,  dx'\     "1 

2  P  {x'  cos  p  —  1/  cos  a)  =  2  m  Ix'  •  -~-   —  1/'  —i—)  ; 


1.  P  [z'  cos  a  —  x'  cos  y )  =  2  m  {  z' 


dt 
dz' 


dt 
dt/  ' 


P  {y'  cos  y  ~  z'  cos  /3)  =  2  in  ly'  •  —^  —  z'  •  -j-  )  ; 


(227) 


174 


ELEMENTS  OF  ANALYTICAL  MECHANICS. 


whence,  the  motion  of  the  body  about  its  centi-e  of  inertia  will  be 
the  same  whether  that  point  be  at  rest  or  in  motion,  its  co-ordinates 
having   disappeared   entirely  from  the  equations. 


ANGULAR   VELOCITT. 

§  174. — Replacing  the  first  members  of  Eqs.  (227)  by  L..  M^,  and  iV^, 
respectively,  §  162  ;  and  substituting  in  the  second  members  fur  dx\  dy' 
and  ds\  their  values  in  Eqs.  (190),  we  readily  find 


d:p 
It 

d^ 
~dt 

dvi 
~dt 


Z.-M;  mx'z 


d-a 


.,.,  ^^^  ^ 


dt  -^         dt 


dt  dt 


(228j 


d\. 

~di 


iV]  +  2  m  .r'y'  •  -Vr  4-  2  m  x  z' 


dt 


If  the  axes  be  principal,  then  will  2  m  x  y'  =  0,  2  m  y'z'  =  0, 
2  m  x'z'^Q;  or  if  the  axes  be  fixed  in  succession,  then  for  the  axis  x'  will 
dib  —  Q\  d(^  —  Q\  for  the  axis  ?/,  c^ip  =  0;  rfw  =  0;  and  for  the  axis 
,?,  (i'cj  =  0;  dip  =:  0,  and  the  above  become 


I^, 


^ 


dt  ~  2m.  (.c'2  -f  y'2) 

d^  _ M, 

dt  ~  ^m  .  (/■'  +  s'2) 

d^  _  iV/ 

Jl  ~  2  «i  .  (y'2  +  s'2)  ■  J 


(22!)) 


That  is,  the  comjwnent  angular  velocity  about  either  a  principal  or  fi.xvil 

axis,  is  equal  to  the  moment  of  the  impressed    fi)rces  divided   by   the 

moment  of  inertia  with  reference  to  that  axis. 

d.s 
The    resultant    ancular    velocitv    bc^ng    denoted    bv     -—-,     we    alsd 
D  ^  a  "      dt 

have,   (Eq.   19G), 


ds 
d 


^  =  ^  X  A9M-  f^+2  +  ^^' 
t  d  t   \ 


(23U) 


MECHANICS    OF    SOLIDS. 


175 


§  175  — The   axis   of  instantaneous   rotation    is    found  as    iu    §  ISS, 
by  making  in  Equations  (190), 

dx'  =  Q;  di/  =zO;  dz'  =  0; 
which  gives, 


r 


.>  ^^ 


\  y  —  X 


,    d\ 


(2311 


d-\,  '       d-m'  ^         ~      d-m 

which  are  the  equations  of  a  right  line  through  the  centre  of  inertia. 


AXIS   OF   SPONTANEOUS   EOTATION. 
176. — If  both  members  of  Equations  (34)  be  divided  by  d  t,  we 


have 


d  X  d  Xj  d  x' . 

d  t  dt  d  t 

dj_  ^  d.y,  dy'  ,^ 

dt  dt  dt  ' 


d  z       ^z,          d  z\ 
dt          d  t           dt 

and  if  for  any 

series  of  elements  we  have 

^•^=0;    ^^   =0;     ^^^    =0; 
dt            '    dt            '    dt            ' 

.     .     . 

then  will 

d  X, 

d  x' .    dy^              d  y'  ,    d  z. 

dz' 

dt 

d  t  ^     d  t    ~          dt   '    dt    ~ 

dt 

(232) 


(233) 


cl  X       cl  ?/  d  X 

and     substitutino;     for   — ^,     -i —     and    —z — ,     their  values    ffivcn    in 
=              dt        dt                 dt  ^ 

Equations  (190),  and  for  -— -j    -y— '     and  -—■>   their  values   given   by 


Equations  (224),  we  find 


dcp 


V .  cos  a      ^ 


•^     c?4.  d  4^ 

dt 
,    dcp  V.QOsb 

a  "d  M  TO 

d  t' 
,    d-^         F.cosc 


(234) 


1/  =  x 


d-7i 


d-us 


I7G  ELEMENTS     OF     ANALYTICAL    MECHANICS. 

Which  are  the  equations  of  a  right  line  parallel,  Equations  (231),  to 
the  instantaneous  axis. 

This  line  is  called  the  axis  of  spontaneous  rotation;  because,  being 
at  rest,  Equations  (232),  while  the  centre  of  inertia  is  in  motion,  tlie 
whole  body  may  be  regarded,  during  impact,  as  rotating  about  this 
line.  Its  position  results  from  the  conditions  of  Equations  (233), 
which  are,  that  the  velocity  of  each  of  its  points,  and  that  of  the 
centre  of  inertia  must  be  equal  and  in  contrary  directions.  The  dis- 
tinction between  the  axes  of  instantaneous  and  of  spontaneous  rota- 
tion is,  that  the  former  is  in  motion  with  the  centre  of  inertia,  wliile 
the  latter  is  at  rest. 

"The  equations  of  the  line  of  the  resultant  impact  are,  Eqs.  (45), 
,  _  Z       ,    ,M,  ,  _Y       ,       L] 

and  the  inclination   0,  of  this  line  to  tht-  spontaneous  axis,  is  given  by 


cos  6 


9 
.Z+  d4,.Y+  da.X 


Vdf'  +  dip'  +  d^^s/z-'\  Y'-{-X'^ 


oi',  substituting  for  c/0,  c/i/),  and  dCo  tlieir  values,  Eqs.  (229),  and  employ- 
ing the  notation  of  Eqs.  (201), 

L,.Z        M,.Y       N,.X 

.      .     (235). 


i/(|)+(f)+(^)-^--M:^'+^ 


The  point  in  which  the  line  of  the  impact  pierces  the  plane  yz  is  given  by 
,       ^^.  ._       A 

dividing  one  by  the  other,  we  have,  for  the  equation  of  the  line  through 
this  point  and  th(3  centre  of  inertia, 

Denote  the  angle  which  this  line  makes  with  the  spontaneous  axis  by  6' \ 
then  from  the  equations  of  these  lines  will 


MECHANICS     OF    SOLIDS.  177 


/c^i+c-:)+>4/(-i;)'+> 

or  Eqs.  (229), 


W 


/(f  ■!)+(!  l)*+'V|-:+' 


(236). 


V  COS  h  V  COS  c 

and  writing  e^  and  <?y  for  • — — —  and  — r^^^'^  respectively,  in  Eqs.  (234), 

Ji  I't 

and  denoting  the  sliortcst  distance  between   the  line    of   the  impact  and 
spontaneous  axis  by  l,  \ve  find  from  the  equations  of  these  Hnes 


('.-f)(S-Tv)  +  ('.-  +  §)(g-|) 


»-/(^:)+{:f:)+'-i/(i-)+(i:)^ 


+1 


1=  = — ._  ~~-^ .    (237). 

sin  0  >Jd<i,-  +  (Z.^^  +  d&'.'VX''  +  y-"  +  Z2 

Make  the  impact  parallel  to  the  axis  x\  then  will  Y  =  0,  Z  =r  0,  and 
N^  =  0,  which,  in  Eq.  (235),  give  cos  0  =  0,  or  6=  90°.  That  is,  if  a 
body  be  struck  in  a  direction  peipendicular  to  the  plane  of  two  of  its 
principal  axes  through  the  centre  of  inertia,  the  spontaneous  axis  will  be 
perpendicular  to  the  line  of  the  impact  and,  Eqs.  (234),  will  lie  in  the 
plane  of  these  axes.  And  if  i?  =  C;  then,  Eq.  (23G),  will  cos  6'  =  0, 
or  0'  =  90°,  and  the  spontaneous  axis  will  be  perpendicular  to  the  line 
drawn  from  the  centre  of  inertia,  normal  to  and  intersecting  the  line  of 
the  impact.  This  latter  will  be  equally  true,  if  the  line  of  the  impact, 
in  addition  to  being  perpendicular  to  the  plane  of  two  of  the  principal 
axes,  also  lie  in  the  plane  of  either  of  these  axes  and  the  third  axis; 
for,  take  the  line  of  the  impact  in  the  plane  xy^  then  Avill  M,  =  0,  the 
denominator  of  Eq.  (230)  becomes  infinity,  and,  therefore,  cos  d'  =  0, 
or  d'  =  90°,  and  this  without  the  equality  of  B  and  C,  Taking  this  last 
position  for  the  line  of  the  impact,  then  will  the  spontaneous  axis  be  in 


ITS  ELEMENTS     OF    ANALYTICAL    MECHANICS. 

the  plane  yz\  F=  0,  Z  =  0,  N^  =  0,  M^  =  0,  sin  0  =  1,  and,  Eqs.  (229), 
dC)  —  0,  dip  =  0,  and,  Eq.  (47),  E  =  X.     These  vahies,  in  Eq.  (237),  give 

Substituting  for  Z^  its  vahie,  Eqs.  (229);  for  H  its  value  M  V,  for 
I,m(x'^  +  y^)  its  value  i/A'/,  and  recalling  that  the  angular  velocity 
about  the  spontaneous  and  instantaneous  axes  are  equal,  and  that  the 
former  is  at  rest, 


'  dt' 


and  dropping  the  subscript  y 


k;     j  e'  +  ^_/  J 


l  =  e  +  -^^ — ■ — '- J    .     .\     .     (238). 

§  177. — The  body  being  free,  and  the  axis  of  spontaneous  rotation  at 
rest,  while  the  other  parts  of  the  body  are  acquiring  motion,  it  is  plain 
that  the  forces,  both  extraneous  and  of  inertia,  are  so  balanced  about  that 
line  as  to  impress  no  action  upon  it.  The  points  of  a  body  on  the  line 
of  the  resultant  impulse  are  called  centres  of  ^^^^'cussion,  in  reference  to 
the  spontaneous  axis.  A  centre  of  percussion  in  reference  to  an  axis  is, 
therefore,  any  point  at  which  a  body  may  be  struck  without  communi- 
cating a  shock  to  a  physical  line  coincident  in  position  with  that  axis. 

§  178. — Denote  by  r'  the  distance  of  an  element  m'  from  the  axis  of 
instantaneous  rotation,  and  by  v'  its  velocity.  Take  the  axis  of  s'  to  coin- 
cide with  the  instantaneous  axis,  and  denote  b)'  a'  and  /3'  the  inclination 
of  the  direction  of  m"s  motion  to  the  axes  x'  and  y',  respectively ;  then  will 

,  ,  ,  ,    (^fp  '      y' .  o'      ^' 

m  V  =  m  r  •  —z— ;  cos  a  =  -, ;  cos  p  =  — - ; 

dt'  r"  /•" 

the  components  of  m'  v',  in  the  directions  of  the  axes  x'  and  y'  are 

m  y  •  -—-,  and         ??i  x  •  -— ; 

^      di'  dt' 

their  moments,  with  reference  to  the  axes  y'  and  .r',  respectively, 
and  the  sums 


m  y  z  •  -r- ;  m  x  z         ■  , 

^         dt  dt' 


dd>    ^     ,    ,  ,  1         d(h 

~r-  •  2im  y  z ,  and          ---  •  zin  x  z . 

dt             ^     '  dt 

If  the  axis  z'  is  principal;    then  2  ??i'?/'a:' =  0,  and  'Lm'x'z'  =  (i\ 


MECHANICS     OF    SOLIDS. 


179 


there  will  be  nothing  to  turn  the  body  about  the  axis  y'  or  x\  ;ind  Vu- 

instantaneous   axis  will    preserve    its    direction   unchang-ed.     If,  thei-el'o;c, 

the    impressed    force    be    so    applied    as    to    cause  the   body  to  begin  to 

rotate  about  a  principal  axis,  the  rotation  will    continue   about  this  axis, 

and    the    axis    is    said    to    be   permanent;    otherwise  the  axis  of  mtation 

will  change  its  position  under  the  pressure    of   the    forces  of  inertia,  till    ■    /  ^J^ 

it  reaches  that  of  a  principal  axis.  <  Urt^  Yt^-  ^i^c^/^*^'-    ^"-'^  '^L.^     \>v4i^ 

MOTION   OF   A   SYSTEM   OF   BODIES. 


§1*79 — We  have  seen  that  the  Equations  (117)  and  (HO)  give 
all  the  circumstances  of  motion  of  the  centime  of  inertia  of  a  single 
body  in  reference  to  any  assumed  point  taken  as  an  origin  of  co- 
ordinates. For  a  second,  third,  and  indeed  any  number  of  bodies, 
referred  to  the  same  origin,  we  would  have  similar  equations,  the 
only  difference  being  in  the  values  of  the  co-ordinates,  of  the  inten- 
sities and  directions  of  the  forces,  and  of  the  magnitudes  of  the  masses. 
This  difference  being  ♦ndicated  in  the  usual  w\ay  by  accents,  we  should 
obtain   by   addition, 


2  M- 


2  M- 


2  M- 


CP'X 


2X: 


=  2F;    I 


2  Z 


(239) 


2  M  \x  •       •' 


2  M  {z 


dfi 
d"^  z 


d"^  x^ 

dt~ 


d'^z 
~d 


0 


^(Yx-Xy); 
^  {Xz  -  Z  x)  ; 


S)  =  ^(^^-^-^); 


(240) 


in  which  it  must  be  recollected  that  x,  y,  s,  &c.,  denote  the  co 
ordinates  of  the  centres  of  inertia  of  the  several  masses  M,  6zG» 
referred    to    a   fixed  oricrin. 


ISO 


ELEMENTS  Of  ANALYTICAL  MECHANICS. 


MOTION    OF    THE    CENTRE    OF   INERTIA    OF   TIIE    SYSTEM. 

§  1 80. — Taking  a  movable  origin  at  the  centre  of  inertia  of  the 
entire  system,  denoting  the  co  ordinates  of  this  point  referred  to 
the  fixed  origin  by  x^,  y^ ,  2^ ,  and  the  co-ordinates  of  the  centres 
of  inertia  of  the  several  masses  referred  to  the  movable  origin  by 
x\  ij\  z\  &c.,  "\ve  have,  the  axes  of  the  same  name  in  the  two  sys- 
tems being  parallel, 

X  —  x^  -\-  x\ 

V  —  y,  +  y\ 

Z  —  s    +  z\ 


and, 


(241) 


(P  X  =^  d"  x^  +  (/"  x\ 

d'^y  —  d'^y,  +  f^-y'r 

f/2  Z   ^(T^Z,^   <P  z\ 

which  substituted   in  Equations  (239),  and  reducing  by   the   relations. 
2j/.c^2.i-'  =  0;     ^Md?xj'  ^^;     ^McPz'  =  Q;    •    .(242) 
obtained  from    the   property  of  the    centre  of  inertia,  we  find 
<Px. 


dfi 

2  J/  = 

2X-, 

cPy. 

dp 

2j/  = 

2F5 

d^z, 
di^ 

2  J/  = 

2Z; 

(243) 


which  being  wholly  independent  of  the  relative  positions  of  the  several 
bodies,  show  that  the  motion  of  the  centre  of  inertia  of  the  system 
will  be  the  same  as  though  its  entire  mass  were  concentrated  in 
that  point,  and   the  forces  applied  directly  to  it. 


§  181. — Multiplying  the  first  of  Equations,  (243),  by  y^,  the  second 


MECHANICS     OF     SOLIDS. 


181 


by  ;r^,  and  taking  the  diffl'vence ;  also,  their  first  by  z^  the  third 
by  x^^  and  taking  the  diftl-rence,  and  again  the  second  by  z^,  the 
third    by  y^,    and   taking    the   difference,    we   find 


\-  (2") 


(^- 


f/2 


a,^    •v5^)-^J/=y,-^2-.-,.^F; 


which  will  make  known  the    circumstances  of  motion  of  the  common 
centre    of  inertia   about   the  fixed   origin. 

iXOTION    OF   THE    SYSTEM   ABOUT   ITS   COMMON   CENTEE   OP   INERTIA. 

§  182. — Substituting  the  values  of  ar,  y,  2,  dr  x,  &c.,  given  by 
Equations  (241),  in  Equations  (240)  and  reducing  by  Equations  (244) 
and    (242),    there   will    result 


2  if. 


2  J/. 


2  J/. 


(. 


cPx' 
d(^ 

rf2  z' 

d  fi 


-  y 


^V-  r'\  1 

-,)=2(F..'-X/) 
0=2  {Xz'  -  Zx') 


d^ "' 

dt 


cPy' 
d  f 


)   =2(Z 


y 


Yz') 


(245) 


Ec|uations  from  which  all  traces  of  the  position  of  the  centre  of 
inertia  have  disappeared,  and  fmm  which  we  conclude  that  the 
motion  of  the  elements  of  the  system  about  that  point  will  be  the 
same,  whether  it  be  at  rest  or  in  inotion.  These  equations  are 
identical  in  form  with  Equations  (118);  whence  we  conclude  that 
the  molecular  forces  disappear  from  the  latter,  and  cannot,  there- 
fore, have  any  influence  upon  the  motion  due  to  the  action  of  the 
extraneous  forces. 


COXSERYATION   OF   THE   AMOTION   OF   THE  CENTRE    OF   INERTIA. 

§  183. — If  the  system  be  subjected  only  to  the  forces  arising  from 
the    mutual    attractions    or    repulsions  of  its    sevei-al    parts,    then    will 

2  X  =  0 ;  2  I"  =  0  ;   2  Z  =  0. 


1S2  ELEMENTS     OF     AX  ALT  TIC  AL     MECHANICS. 

Fur,  the  action  of  the  mass  M,  upon  a  single  element  of  iLT, 
will  vary  with  the  number  of  acting  elements  contained  in  M ', 
and  the  effort  necessary  to  prevent  M'  from  moving  under  this 
action  will  be  equal  to  the  whole  action  of  AI  upon  a  single  element 
of  M'  repeated  as  many  times  as  there  are  elements  in  M'  acted 
upon  ;  whence,  the  action  of  M  upon  M'  will  vary  as  the  product 
MM'.  In  the  same  way  it  will  appear  that  the  force  required  to 
prevent  M  from  moving  under  the  action  of  J/',  will  be  propor- 
tional to  the  same  product,  and  as  these  reciprocal  actions  are 
exerted  at  the  same  distance,  they  must  be  equal ;  and,  acting  in 
contrary  directions,  the  cosines  of  the  angles  their  directions  make 
with  the  co-ordinate  axes,  will  be  equal,  v^^ith  contrary  signs.  Whence, 
for  every  set  of  components  P  cos  a,  P  cos  /3,  P  cos  y,  in  the 
values  of  2  X,  2  F,  2  Z,  there  will  be  the  numerically  equal  com- 
ponents, —  P'  cos  c//,  —  P'cos/3',  —  P'  cos  y'i  <ind,  Equations  (243), 
reduce,    after    dividing   by    2  M,    to 

■^-^'    -d¥  -^^    Ifi-^'    '    '    '    ^-^^^ 

and    from  which  we    obtain,  after    two   integrations, 

x^=   C'.t  +  J)';     -] 

y,  ^   C"J  +  I)"-     \ (247) 

z,  =   C"'.t  +  I)'" -J 


m  which  C",  C",  C",  D\  D"  and  D'"  arc  the  constants  of  inte- 
gration ;  and  from  which,  by  eliminating  /,  we  lind  two  equations  of 
the  lirst  degree  between  the  variables  x^ ,  y^ ,  z^ ,  whence  the  path 
of  the    centre    of  inertia,  if  it   have   any  at    all,   is    a    right    line. 

Also    multiplying    Equations  (246)   by   2dXj,  2di/j,  2d2^,  respec- 
tively, adding  and    integrating,  wc   have 

'll±J^jlfl  =  K=  =  . (...) 

in  which  C  is  the  constant  of  integration  and  V  the  velocity  of  the 
centre  of  inertia  of  the  system.  Eroni  all  of  which  we  conclude, 
that    when   a   system    of    bodies    is    subjected    only  to    forces    arising 


MECHANICS     OF     SOLIDS. 


183 


from  the  action  of  its  elements  upon  each  other,  its  centre  of  inertia 
will  either  be  at  rest  or  move  uniformly  in  a  right  line.  This  is 
called   the   conservation  of  the  motion  of  the   centre  of  inertia. 


CONSEKVATION     OF    AEEAS. 

§184. — The    second   member    of  the  first  of  Equations  (-4.3)  may 
be  written, 

Yx'  -  Xy'  +  ¥'  x"  -  X'y"  +  &c. ; 
and    considering   the   bodies   by  pairs,  we  have 

A"  =  -  X' ;     F  =  -  F' ; 
and  eliminating  X'  and   Y'  above  by  these  values,  we   have 

Y{x'  -  x")  -  Xiy'  -  y")  +  &c. 
But, 


P  P 

in  which  p  denotes  the   distance   between    the   centres    of    inertia   oi 
the  two  bodies.     And  substituting  these  above,  we  get 
?/'  —  v"  r'  —  t" 

and  the   same   being   true   of  every    other  pair,  the  second   members 
of  Equations   (245),  will    be  zero,  and    we   have 


2.¥. 


2  J/. 


(= 


^'•S')-«= 


If 

d?x' 

~d¥       '"      dP 


dt 
(/2  z'^ 


V      di^  dfiJ 


and  integrating 


x'  dy^  y'  d  x'  _  ^ 

jt      -  ^  '  ; 


t^vfeV    «*^  Oi/\,^  u*~^ ,  z'  d  x'  —  x'  d  z' 


I  ir 


(249) 


,,J_^.*K^,>v^^   *ir\\.   y'  d  z'  -  s'  d  y'  _ 


/Mft. 


.  »N_  I  _  ,   r     /-u 


184:    ELEMENTS  OF  ANALYTICAL  MECHANICS. 

But  §  190,  x'  dy' —  y'  tl  x[  is  twice  the  differential  of  the  area  swept 
over  by  the  projection  of  the  radius  vector  of  the  body  M,  on  the 
co-ordinate  plane  x'  y\  and  the  same  of  the  similar  expressions  in 
the  other  equations,  in  reference  to  the  other  co-ordinate  planes ; 
whence,  denoting  by  A^,  A  ,  ^l^.,  double  the  areas  described  in  any 
interval  of  time,  t,  by  the  projections  of  the  radius  vector  of  the  body 
M,  on  the  co-ordinate  planes,  x'  y\  x'  z\  and  y'  z' ,  and  adopting 
similar  notations  for  the  othei*  bodies,  we  have 

d  A, 
dt  ' 

dt 

dt 

in  which  C",  C",  C"\  denote  the  sums  of  the  products  obtained  by 
multiplying  each  mass  into  twice  the  area  swept  over  in  a  unit  of  time 
by  the  projection  of  its  radius  vector  on  the  {ilaiies  x'  y\  x'  z\  y'  s' ;  and 
by  integrating  between  the  limits  t^  and  t\  giving  an  interval  equal  to  i, 

:^M.A^  =  C'.t; 
-LM  .Ay  =  C"  t; 
2  M.  A^  =   C"  t; 

whence  we  find  that  when  a  system  is  in  motion  and  is  only  sub- 
jected to  the  attractions  or  repulsions  of  its  several  elements  upon 
each  other,  the  sum  of  the  products  arising  from  multiplying  the 
mass  of  each  element  by  the  projection,  on  any  plane,  of  the  area 
swept  over  by  the  I'adius  vector  of  this  element,  measured  from 
the  cpntre  of  inertia  of  the  entire  system,  varies  as  the  time  of  the 
motion.     This  is  called    the  principle    of  the   conservation  of  areas. 

§  185. — It  is  important  to  remark  that  the  same  conclusions 
would  be  true  if  the  bodies  had  been  subjected  to  forces  directed 
towards  a  fixed  point.  For,  this  point  being  assumed  as  the  origin 
of  co-ordinates,  the  equation  of  the  direction  of  any  one  force,  say 
that   acting   upon  M,  will   be 

Yx-  Xy  =  0; 


MECHANICS     OF     SOLIDS.  IS5 

and  the  second  members  of  Equations  (240)  M'ili  reduce  to  zero ; 
and  the  form  of  these  equations  behig  the  same  as  Equations  (245), 
they  will   give,    by  integration,  the    same   consequences. 

INVAKIABLE    PLANE. 

d  y' 

§  186. — If  Ave  examine  Equations  (249),  we  shall  find  that  M-  —j^ 

is  the    quantity  of  motion   of  the    mass    J/,    in    the  direction  of  the 

axis    ?/',  and   is  the   measure  of  the  component  of  the    moving  force 

dx' 
in   that  direction  ;    the   same   may  be  said    of  JM-  —, — i    in  the  direo- 
'  -^  di 

tion    of  the   axis   x' ;  whence   the    expression, 
,^  x'd  if  —  y' d  x' 

M'     ^ ,^ 5 

dt 

is  the  moment  of  the  moving  force  of  M^  with  respect  to  the 
axis  z' .  Designating,  as  before,  the  sum  of  the  moments  with  respect 
to  the  axes  2',  y'  and  .r',  by  L^ ,  M^ ,  N^ ,  respectively,  Equations  (249) 
become 

Z,  =  C"  ;     M,  =  C"  ;     N,  =  C". 

Denoting   by    6^,  G^,  and    0;^,  the   angles    which   the   resultant   axis 
makes    with   the   axes  2',  y'  and  x\  we   have,  §110, 

cos  6,  =   =z=:==^=^=:     =    : 


M,  C" 

N.  C" 

cos  Qx  = 


These  determine  the  position  of  the  resultant  or  principal  axis. 
The  plane  at  right  angles  to  this  axis  is  called  the  principal  plane. 
The  position  of  this  plane  is  invariable,  and  it  is  therefore  called 
the  invariable  plane,  either  when  the  only  forces  of  the  system  are 
those  arising  from  the  mutual  actions  and  reactions  of  the  bodies 
upon  each  other,  or  when  the  forces  are  all  directed  towards  a  fixed 
centre. 


186  ELEMENTS    OF    ANALYTICAL    MECHANICS. 

^  / 

"  '  PRINCIPLE    OF    LIVING    FORCE. 

§  187. — If,  during  the  motion,  two  or  more  bodies  of  the  system 
impinge  against  each  other  so  as  to  produce  a  sudden  change  in  their 
velocities,  the  sum  of  the  living  forces  Aviil  undergo  a  change.  To  esti- 
mate this  change,  let  A,  B^  G  be  the  velocities  of  the  mass  ?n,  in  the 
direction  of  the  axes  before  the  impact,  and  a,  b,  c  what  these  veloci- 
ties become  at  the  instant  of  nearest  approach  of  the  centres  of 
inertia  of  the  imjoinging  masses,  then  will 

A  —  a^    Jj  —  b,     0  —  c, 

be  the  components  of  the  velocities  lost  or  gained  hj  m  at  the  instant 
corresponding   to   this   state  of  the   impact,  and 

m  {A  —  a),     m  {B  —  b),    m  {0  —  c), 

the  components  of  the  forces  lost  or  gained.  The  same  expressions, 
•with  accents,  will  represent  the  components  of  the  forces  lost  or 
gained  by  the  other  impinging  bodies  of  the  system.  These,  by 
the   principle    of  D'Alembert,  §  Tl,  are  in  cquilibrio,  whence 


2  m 


{A  —  a)  6  X  -i-  H  m  {B  ~  b)  Sy  +  I.  m  {C  —  c)  o  z  =  0. 


The  indefinitely  small  displacements  8 x,  St/,  o z,  d:c.,  must  be  made 
consistently  with  the  connection  by  virtue  of  which  the  velocities  are 
lost  or  gained;  but  as  a,  b,  c  denote  the  components  of  the  actual 
velocities  of  any  two  bodies  during  the  time  of  greatest  compression 
when  alone  these  velocities  are  equal,  this  condition  will  be  fulfilled 
if  we  make 

S  X  =:  a.S  t;      S  y  _=  b  .0  t;      o  z  ^  c  .  S  i. 

These  values   being    substituted   in    the   above   equation,  we   have, 
after   dividing   by  S  t, 

2  m  (A  ~  a)u  +  i:  m  {B  —  h)  b  +  1  m  {C  —  c)  c  =  0  •  •   (251) 
or, 

2?«  {A  a  4-  Bb  +  Cc)  -  Zm  {a?  +  ^2  -f-  c^)  =  0    •    •  (252) 


MECHANICS     OF     SOLIDS  IS^ 

But   Ave   have   the   identical    equation, 

L  -f-  c-  —  •2[Aa  -\-  Jjb  +  Cc)^ 
or, 

,.j2  ^    2,>2    ^    (72  a2    4.    ^,2    .^    ^2 


I         {A-aY  +{B-bY  +  {C-c)~ 
L  2 

which  iu  Equation  (252)  gives, 

1  m{A~-{-£'~-\-  C~)-:e  ?n{cr-  +  b'  +  c-)=l7a[{A-a)"  +  {Il-iy--\-{C~cy], 

aud   making 

A^  +  ^2  ^  (72  ^  72^ 
rt^    -j_    ^2    _j_    g2    _  ^2^ 
2  m  F^  —  2  m  II-  =  2  »i  [  (.1  —  a)-  +  {B  -  h)'^  +  {C  —  c)2]  •  •   (2.53) 

whence  we  conclude,  that  the  difference  of  the  sums  of  the  living 
forces  before  the  collision,  and  at  the  instant  of  greatest  compression. 
Id  equal  to  the  sum,  of  the  living  forces  which  the  system  would  have, 
if  the  masses  moved  with  the  velocities  lost  and  gained  at  this  stage 
of  the    collision. 

Since  all  the  terms  of  the  preceding  equation  are  essentially 
positive,  it  follows  that  at  the  instant  of  nearest  approach  of  the 
impinging  bodies,  there  is   a   loss   of  living    force. 

If  the  impinging  masses  now  react  upon  each  other  in  a  way  to 
cause  them  to  be  thrown  asunder,  and  A',  B\  C\  (Szc,  denote  the 
components  of  the  actual  velocities,  in  the  direction  of  the  axes,  at 
the  -instant  of  separation,  then  will  the  components  of  the  velocitic3 
lost   and  gained   while    the   separation   is  taking   place,  be 

a  —  A',     h  -  B\     c  -  C",    (kc,  &e.  ; 

and   Equation    (251)  will   become 

2  m  (a  -  ^1')  a  +  2  ??j  (6  —  i?')  i  +  2  m  (c  —  C')c  —  0, 
or. 

2m  (rt2  4-  i2  4.  c2)  -  2?n  {A' a  +  B' b  +   C c)  -  0  ; 


18S 


ELEMENTS     OF    ANALYTICAL    MECHANICS. 


(a  -  Ay  +  {b  -  By  +  {c  -  C'Y 


and    eliminating  A!  a  -\-  B' h  -\-  C  c^  by   means  of  the    identical   equa- 
tion, 

d^  +  /,2  j^  c^  +  A'-^  +  i?'2 
+  C"'--2{A'a  +  B'h  +  C'c), 
we   obtain, 

{  +  {c-  C'f  J 
and  making 

^.'2    _|.    ^'2.   ^_     (7'2    ^     p2^ 

2  ??i  m2  _  2  m  P^'3  3=  -  2  wi  [  (a  -  .4')2  +  [h  -  By  +  {c  -  Cy\  ■  •  ('Jo4) 

All  the  terms  of  this  equation  being  essentially  positive,  it  fol- 
lows, from  the  sign  of  the  second  member,  that  during  the  reaction 
of  the  bodies  by  which  they  are  separated,  there  is  a  gain  of  living 
force. 

If  the  loss  and  gain  of  velocities  after,  be  the  same  as  before 
the  instant  of  greatest  compression,  then  will  there  be  no  loss  or 
gain    of  living    force   by  the    collision. 

PLANETARY     MOTIONS. 


§  188. — When  the  only  forces  are  those  arising  from  the  mutual 
attractions  of  the  several  bodies  of  the  system  for  one  another,  the  sec- 
ond members  of  Equations  (239)  reduce,  as  we  have  seen,  §  183,  to  zero, 
and  those  equations  become 


(2.55) 


2  M.  —  =  0, 
a  t 


jct  us  now  find  tlie  motion  of  any  one  body  of  the  system  in  rei'cr- 
';nce  to  any  othei',  taken  at  pleasure.  Tliis  latter  body  will  be  called 
the   central^   the    former    the    'primarij^  and    the    others,  collectively,   the 


M  E  C  11  A  X  IC  S     O  F    S  0  L I  D  S  .  1  ■  <» 

•pcrturbatiny  bodies.  Let  the  central  and  j^wimaiy  bodies  be  those  whose 
nlas^^•s  are  J/ and  M,  respectively;  the  pertiirbating  bodies  those  whose 
masses  are  J/^^,  -'j^,,,,  itc.  The  tirst  of  the  above  eqnations  iiiav  be 
written 

^^   d^  X        ,,    d'- .V  ,,      d'x,, 

''■i^  +  "'-T¥  +  ^""-irf-''  ■  ■  ■  ^''>> 

If  the  pcrtr.rbalinL!,'  bodies  alone  acted  upon  one  another,  the  last  term 
would  be  zero;  and  when  the  action  of  the  central  and  primary  are 
included,  the  numerical  value  of  this  term  will  result  from  the  action 
of  these  latter  bodies..  Denote  the  reciprocal  action  of  any  two  bodies 
upon  one  anotlier  by  -writing-  their  masses  within  the  jiarenthetic  sign, 
and  use  the  subscript  x  to  denote  the  component  of  this  action  parallel 
to  the  axis  x.     Then  ^vill 

2  (J/  3/ J.  +  2  (.1/  J/ ,  ).  -  2  J\l^  ^  =  0  ; 

adding  this  to  the  next  equation  above,  we  get 

^^-  !77  +  ^^^-  T?  +  -  ^^^"^^"^^  +  ^  ^^^'  ^^"^'  ==  ^  •  •  (^^^) 

Taking  the  movable  origin  at  the  centre  of  the  body  J/,  we  have 

Xj  =  X  —  x',  and  d^  x^  —  d'-  x  —  d^  .r', 
which,  substituted  above,  gives 

(^^^  +  K)  "^r^  -  ^^^.  •  '^'  +  ^  i^^^^Kh  +  2  (i/  i/J.  =  0  ; 
diviiliug  by  il/+ J/    and  multiplying  by  J/,  there  will  result 
d'x      M.M,    d'x'  M  ._„  .  M 


M. 


^^  -  iZTlZ/ 77  +  .¥T^  -  (^^^^^^'')' +i7T]i^^ 


The  value  of   the    first    term    results    from   the  component  action  of 
the  primary  and  perturbating  bodies  upon  J/;    whence 

^^:  .77  -  W^'^^')''  -  -  (^^^^^^")J  =  0 ' 

from   which  subtiacting  the  equation  above,  tliere  Avill  result 


190 


E  L  E  M  E  N  T  S     OF     A  X  A  L  Y  TI  C  A  L     M  E  (J  II  A  N  1  C  S  . 


Dividing-  by  the  cojiHcieiit  of  the  first  term,  and  treating  the  other  two 
of  E(|iiat:on.s  {2oo)   in  tlie  same  way,  we  finally  get 


Tf  -  JTji'  ■  ^'"■''^■+  Tr  -  (■'^-'^")-  -  i  •  ^  <^^-  ^'^")-  =  "' 


d'y'       M+M^ 
~d¥  ~  M .  M^ 

d'z'      M+M^ 


1 


V7-l/V-(^^^^)^  +  ^-^(^^^^)^-i7 


;(j/,.]/j,.=  o, 


\  (258) 


M .  M  ■  ^^^^'^'-  -^Tl-  ^^^^'^^  -  i  •  ^  (^^'  ^^"^'  =  '• 


df       M.M  '^'"  "'■  ■  M 

Which,  by  integration,  will  give  all  the  circumstances  of  the  j)rimary's 

motion  in  reference  to  the  central  body. 


LAWS    OF    CENTRAL    FORCES. 


§  189. — A  central  force  is  one  which  is  directed  towards  a  centre, 
movable  or  fixed,  and  of  which  th.c  intensity  is  a  function  of  the  dis- 
tance from  the  centre.     Tlie  forces  of  nature  are  of  this  description. 

If  tlie  perturbating  bodies  did  not  exist,  then  would  the  action  on 
the  primary  be  directed  to  the  central  body  as  a  centre,  the  Equations 
(258)  wovild  reduce  to  their  fii'st  two  terms,  and,  denoting  the  distance 
from  the  centi'al  to  the  primary  by  r',  they  would  be  written, 


d'x  M+M^     ,,r.r.  M+M, 

M.M     ^         '         M.M, 


dt' 
d'y' 


di 

cZ-  z  _  M^M, 

HF^lM.M. 


M^-M,    .^^^^         M+M^ 

MVW/^'''''^-'=-jrw^ 


{M.M^} 


M+M^ 
'  M.M, 


{MM) 


{MM) 


{MM)."-. 


(250) 


J 


Multiply  the  first  by  y',  the  second  by  a:',  and  take  the  difference  of  the 
products;  also  multiply  the  first  by  2',  the  third  by  x\  and  take  the  differ- 
ence of  the  products;  and  again  tlie  second  by  2',  the  ihiivl  by  y',  and  take 
the  difference  of  the  products  :    there  will  result,  omitting  the  accents, 


d'^  1/  d^  X 

d¥'''~d'f''^ 


0, 


d'x 
df 


d'z 

Z r-r,  .  rC  =  0, 


dH 

df-y- 


de 

dr  y 
If' 


2=.0; 


MECHANICS     OF    SOLIDS, 


191 


which, 

being 

intcg 

•atetl,  give 

dy 
dt' 

x 

dx 
~  dt 

y 

= 

C", 

d  X 
dl 

z 

dz 

~Tt 

X 

== 

C", 

dz 
dTt' 

y 

_d_y_ 
dt 

.z 

= 

C'"- 

(260) 


in  which  C,  C'\  and  C"  are  the  constants  of  integration. 

Multiplying  each    by  tlie  first  power  of   the  variable   w'hich  it  does 
not  contain,  and  adding,  we  have 

C'z  +  C"y  +  C"'x  =  0  ;  . 

which  is  the  equation  of  an   invariable  plane  passing  through  the  cen-'     ^  \j^^ 
tre,  and  of  which  the  position   depends  upon  the  constants  C\  C"^  C^".  A.  Ji^^i^ 
Whence  we   conclude   that    the    priniarj^  deflected   by  the   central   body  l(,itli..C 
alone,  will  describe  a  plane  curve  of  which    the   plaue  will   contain    the  ^. 
centres  of  both. 

§  190. — Take  the   co-ordinate   plane  xy  to  coincide  with  this  plaue, 
and  the  Equations  (260)  will  reduce  to 


d  y  d  X 

—-  .x  —  -—  .y  ■znW 

dt  dt   -^ 


(261) 


Transform  to  polar  co-ordinates;  for  this  purpose  we  have 

.r  =  ?• .  cos  a  ;    y  =  ?• .  sin  ct ; 

differentiating, 

d  X  =  d  r  cos  «.  —  r  sin  ad  a, 

d  y  =  d  r  sin  a  -\-  r  gos  ad  a. 
Substituting  in  Equation  (2G1),  we  find 


dy 
dl  "^ 
iutea'ratino;  again,  we  have 


dx 

Tt-y 


dt 


(262) 


fr\da=.C't+  C", 

and  taking  between    the    limits  r^,  a^  and  r^^,  a^^,  corresponding  to  the 
time  t^  and  t^^, 

f^'  r\da^C'{t^-t;) (263) 


1 D-J  !•:  i -  !•;  -M  \i  N  T S     OF     ANALYTICAL    M E C  II A N  I  C  S. 

But  i  rda  is  doiillc  the  area  dcsci'ibed  by  the  motion  of  the  radius 
vector;  whence  we  see,  Equation  (263),  that  the  areas  described  by  the 
radius  vector  of  a  body  levolving-  about  a  centre,  are  proportional  to  the 
intervals  of  time   required  to   describe  them. 

JNIaking,  in  Equation  (203),  t^^  —  t^  equal  to  nnity,  the  first  member 
becomes  double  the  area  described  in  a  unit  of  time.  Denoting  this 
by  2  c,  that  equation  gives 

C  =  2  c. 


Placing  this  in  Equation  (263),  ^ve  find 


r    (I 


r'^ .  (/  a 


'"-'-  =  -^1-^ — (-«^) 

That  is  to  say,  any  interval    of  time  is  equal   to  the  area  described 
in  that  interval,  di\ided  by  the  area  described  in  the  unit  of  time. 

§  191. — The  converse  is  also  true;  for,  dift'erentiating  Equation  (262), 

we  find 

f/°  ri  d^  X 

di^  dt'  -^  ' 

Multiplying  by  Jf,  and  replacing  M .  -~  and  M .  -j-^   by  their  values 

in  Equations  (120),  there  will  result 

Tx  —  Xy  =  0 (2G5) 

wliich  is  the  Equation  of  the  line  of  direction  of  the  force;  and  having 
no  independent  term,  this  line  passes  through  the  centre.  Whence  we 
conclude,  that  a  body  whose  radius  vector  describes  about  any  point 
areas  proportional  to  the  times,  is  acted  upon  by  a  force  of  which  the 
line  of  direction  j^asscs  through  that  point  as  a  centre.  The  force  will 
L'C  attractive  or  repulsive  according  as  the  orbit  turns  its  concave  or 
convex   side  towards  the  centre. 

§  192. — Replacing  C  by  its   value  2  c,  in   Equation   (262),  and  divi- 
ding by  r'',  we  have 

da       2  c 

,77  =  7^ (2«») 


MECHANICS    OF    SOLIDS.  193 

The  first  member  being  the  actual  velocity  of  a  point  on  the  radius 
vector  at  the  distance  unity  from  the  centre,  is  called  the  angular  ve- 
locity of  the  body.  Tlie  angular  velocity  therefore  variis  iniersehj  as 
the  square  of  the  radius  vector. 

§  193. — Multiply  Equation  (26C)  by  d  s,  and  it  may  be  put  undci 
the  form, 

ds  2  c 


d  t  r  d  a'' 

d  s 

but  ~ — ■,  is  equal  to  the  sine  of  the  ano-le  which  the  element  of  the 
d  s 

orbit  makes  "vvith  the  radius  vector,  and  denoting  by  j?  the  length  of 
the  perpendicnlar  from  the  centre  on  the  tangent  to  the  orbit  at  the 
place  of  the  body,  Ave  have 

r .  d  a 
p  =  r.  — — , 
d  s 

and 

V^J (26V) 

whence,  the  actual  velocity  of  the  body  varies  inversely  as  the  distance 
of  the  tangent  to  the  orbit  at  the  body's  place,  from  the  centre. 

§  194. — Denoting  the  intensity  of  the  acceleration  on  3£^  by  i^;  sub- 
stituting M^  .  F.dr  for  Xdx  +  Td?j  -\-  Zds,  writing  M^  for  M  in  the 
coefficient  of  V^  in  Equation  (121),  and  differentiating,  we  find 

VdV^-Fdr- 

and  taking  the  logarithms  of  both  members  of  Equation  (267), 

log  V  =  log  2  c  —  log  p ; 
differentiating, 

d  V  dp 

and  dividing  the  equation  above  by  this, 


TT-a        T-,        dr           —  1        dr  •  ,       . 

V^  =  F.p.--  =2F'.-p.—  .     .     .     (268) 

dp           *\2    _    dp  ^       ' 

13 


:  o- :  /■   ^  /^JO^  =    i2A  =   '^  ^    <-^ 


Xu^ 


:s^  ^  -^ 


U..  .^f' 


r.u 


ELEMENTS    OF    ANALYTICAL    MECHANICS. 


Whence  we  conclude  that,  the 
velocity  of  a  body  at  any  point 
of  its  orbit  is  the  same  as  that 
which  it  would  have  acquired  had 
it  fallen  freely  from  rest  at  that 
point  over  the  distance  M E^  equal 
to  one-fourth  of  the  chord  of  cur- 
vature M  G,  through  the  fixed  cen- 
tre— the  force  retaining  unchanged 
its  intensity  at  M. 

§  195. — Resuming  Equations  (120),  we  have 


dt' 


d  X 

dl 
~dt' 


and  performing  the  operation  indicated,  regarding  the  arc  of  the  orbit 
as  the  independent  variable,  we  have,  after  dividing  both  numerator  and 
denominator  by  d  s^, 


X=M 


dtd^x      d  X   d^  t 

d  s   d  .s-^       d  s    ds^ 

d¥ 


whence. 


In  like  manner, 


di  dH 
Jl'd? 


Lci^ 


d^  X      d  X   d^  s' 


r^2  ^^^  y    '■^  y  ^^  *" 


Squaring  and  adding, 


MECHANICS     OF    SOLIDS. 


195 


^  I  d  (.'  \d  a    d  6-^       d  s    d  s^       d  >i    d  s'/ 

but,  denoting  the  radius  of  curvature  by  p,  we  have 

Us)  '^  \d7)  "^  \^777  ~  ^^ ' 

and  multiplying  the  second   term  of  the  second   member  of  the  prece- 
ding equation  by  -,  it  may  be  put  under  the  form, 

MV   M.d's/dx      dKx      dy      cP y      dz      d 


dt 


rs/dx      d'Kx      dy       cP  y      dz       d'^s\ 
"^  \d~s  ' '^ dV  ^  dl-  '  ^  I?  '^  cTs'  ^^  d7')' 


or, 


MV    M.dU  ,    . 

2  • .  •  cos  6  ;  -^^ 

p  d  V 


in  which  6  denotes  the   angle   made  by  the   element  of  the  curve  and 
radius  of  curvature  ;  also 

d  x"^      d  11^      d  z^ 

d  s'  ^  d  s'  ^  ds'~     ' 

whence,  substituting  for  X^  +  Y^  +  Z^  its  value  H',  we  have 

' '  \d  f)  ' 


M'^V'           MV   M.d's 
R-  = T \-  2  . ■ —-^ —  •  cos  o  -\-  M 


df 

and  comparing  this  with  Equation  (56)  we  find  that  R  is  equal  to  the 
resultant  of  the  two  component  forces 

and  il/--r— „ 

p  d  t^' 

which  make  with  each  other  the  angle  6.     But  6  is  equal  to  90°,  and 
therefore 


p-  \d  t'l 


(269) 


See  Appendix  No.  2. 


196  ELEMENTS     OF    ANALYTICAL     MECHANICS. 

The  second  of  these  components  is,  Equation  (13),  the  intensitv  of 
the  reaction  of  inertia  in  the  direction  of  the  tangent,  and  the  first  is 
therefore  its  reaction  in  the  (Hrection  of  the  i-adius  of  curvature. 

This  first  component  is  called  the  centri/a[/al  force,  and  mav  be  de- 
fined to  be  the  resistance  which  the  inertia  of  a  body  in  motion  opposes 
to  lohatever  deflects  it  from  its  rectilinear  path.  It  is  measured,  Equa- 
tion (209),  by  the  living  force  of  tlie  body  divided  by  t!ie  radius  of 
curvature.  The  direction  of  its  action  is  from  the  centre  of  curvature, 
and  it  th-:.s  differs  from  the  force  Aviiich  acts  towards  a  centre,  and 
whicli  is  called  centripetal  force.     The  two  are  called  central  forces. 

If  the  component  in  the  direction  of  the  orbit  be  zero,  then  will 

d  t' 
and  denoting  the  centrifugal  force  by  F ^ ,  Ave  have 

^.  =  -— (270) 

and  integrating  the  next  to  the  last  equation,  we  have 

d  s 

in  whicii  C  is  the  constant  of  integration.  Whence,  the  velocity  will 
be  constant,  and  wc  conclude  that  a  body  in  motion  and  acted  upon 
by  a  force  whose  direction  is  alwjiys  normal  to  the  path  described,  will 
presen'e  its  velocity  unchanged. 

These  laws,  except  that  expressed  by  Equation  (208),  are  wholly  in- 
dependent of  the  intensity  of  the  extraneous  force  and  of  the  law  of  its 
variation.     Not  so,  however,  of 

THE    ORBIT. 

§  100. — To  find  the  differential-  equation  of  the  orbit,  multiply  the 
first  of  Equations  (259)  by  2  d  x,  the  second  by  2  d  y,  add  and  inte- 
grate;   we  find,  omitting  the  accents, 

dx'  +  dj/"-       3T+Af     f(MM\    2-^'f^^  +  2.yfZy 


MECHANICS     OF    SOLIDS.  11J7 

but 

»•'"  =  x"-  -\-  ir,  and  r  a'  r  =.  a; d x  -\-  y  dy  '^ 

also 

.r  z=  r  cos  a, ;     y  ■=.  r  .  sin  a  ;  1.  >  v-,.^  y 

f/  X  =  —  )•  sin  u.  d.  a  -\~  cos  ad  r  '^        *>  J  K,^i  dt  sc  4»-  '^'^  ^«^ 

d y  =  r  cos  a c? «,  +  sin  adr  \ 


V 

'k' lie  t/ 


and,  Equation     (266), 

1    __     2c 

These  substituted  above,  give 

4„'(^+i^V=2.^4±#'./-(i/J/,),J,.. 
Make  ua-ui>,d^~^^  ^3-^ 

-  =  ?f,  and  tlicrefoi'e    — ■  =  —  cZ  i<,  <a^  -  --i^Uctt  .-  -  ^:?* 

substitute  above,  dift'erentiate  and  reduce,  there  will  result 

and  making 

iT    =     j  ''^  "   '^  +  —,^1  =  relative  acceleration  on  M,    .     (271) 

F  =  4  c' zi"  ■  l^-^l  +  2i\ (272) 

From  which  the  equation  of  the  orbit  may  be  found  by  integration, 
■when  the  law  of  the  force  is  known;  or  the  law  of  the  force  deduced, 
when  the  equation  of  the  orbit  is  given. 

In  the  first  case,  the  integral  will  contain  three  arbitrary  constants 
— two  introduced  in  the  process  of  integration,  and  the  third,  c,  exist- 
ing in  the  differential  equation.  These  are  determined  by  the  initial 
or  other  circumstances  of  the  motion,  viz. :  the  body's  velocity,  its  dis- 
tance from  the  centre,  and  direction  of  the  motion  at  a  given  instant. 
The  general  integral  only  determines  the  nature  of  the  orbit  described : 
the  circumstances  of  the  m  jtion  at  any  given  tiraofdetermine  the  species 
and  dimensions  of  the  orbit. 


f 

19'>  f:i.K.MK?>TS     OF     ANALYTICAL     MECHANICS. 

Ill  the  second  case,  find  the  second  difFercntial  cooilicicnt  or'  n  in 
regard  to  a,  Irom  the  poliu-  equation  of  the  curve ;  substitute  tliis  in 
the  above  equation,  eliminating  a,  if  it  occur,  by  means  of  the  relation 
between  m  and  a,  and  the  result  will  be  F^  in  terms  of  u  alone. 


/^  SYSTEM    OF    THE    WORLD. 

§  197. — The  most  remarkable  system  of  bodies  of  which  wo  have 
any  knowledge,  and  to  which  the  precedmg  principles  have  a  direct 
application,  is  that  called  the  solar  system.  It  consists  of  the  Sun, 
Uu'  Pliinets,  of  which  the  earth  we  inhabit  is  one,  the  Satellites  of  the 
planets,  and  the  Comets.  These  bodies  are  of  great  dimensions,  ai'e 
sj)heroidal  in  figure,  are  separated  by  distances  compared  to  which 
their  diameters  are  almost  insignificant,  and  the  mass  of  the  sun  is 
so  much  greater  than  that  of  the  sum  of  all  the  others,  as  to  bring 
the  common  centre  of  inertia  of  the  whole  within  the  boundary  of 
its  own   volume. 

Thfse  bodies  revolve  about  their  respective  centres  of  inertia,  are 
ever  shifting  their  relative  positions,  and  our  knowledge  of  them  is  the 
result  of  computations  based  upon  data  derived  from  actual  observation, 

Kepltr  found  ;  . 

I.  That  th.e  areas  sivept  oiu  r  by  tlte  radius  vector  of  eff«h  -planet 
ahout  the  sun,  in  "'tl^-SiiMe  orbit,  are  proportional  to  the  times  of  de- 
scribing  them. 

II.  TJiat  the  2yla7icts  move  in  ellipses,  each  having  one  of  its  foci  in 
the  suns  centre. 

III.  That  the  squares  of  the  periodic  times  of  the  jdanets  aJfout  the 
sun,  are  proportional  to  the  cubes  of  their  mean  distances  from  that 
bod)/. 

These  are  called  the  laws  of  Kepler,  and  lead  directly  to  a  knowl- 
edge of  the  nature;  of  the  forces  which  uphold  the  solar  svstcm. 

CONSEQUENCES    OF    KEPLKk's    LAW'S. 

§  198. — Tiie  first  hiw  shows,  §  191,  that  the  centripetal  forces  which 


MECIIANIvJS     OF    SOLIDS.  199^ 

keep  the  planets  in  their  orbits,  are  all  directed   to   the    sun's    centre  ; 
and  that  the  sun  is,  therefore,  the  centre  of  the  sijstem. 

§  199. — What  law  of  the  force  will  cause  a  primary  to  describe 
about  a  central  body  an  ellipse  having  one  of  its  foci  at  the  centre  of 
the  latter  ?     The  equation  of  the  ellipse  referred  to  its  focus  as  a  pole  is 


whence, 

and, 

d}  u        —  e  cos 
J^""  ^  a  (1  ^e^)' 

whicli,  substituted  m  Equation  (2*72),  give 


r 

= 

a  (1  -  e')  ^ 
1  +  e  cos  a  ' 

1 

r 

= 

u 

1  +  e  cos  a 
~~   «  (1  -  e'^)  ' 

cF 

u 

—  e  cos  a 

7-1        i    0    0  /~  ^  cos  a       1  +  e  cos  a\ 

reducing  and  replacing  ti,  by  its  value  -,  we  have 

4-  r'         1 

and  from  which  we  conclude,  that  the   only  law  for  the  relative  accel- 
eration, is  that  of  the  inverse  square  of  the  distance. 

§  200. — Conversely,  let  the  force  vary  inversely  as  the  square  of  the 
distance ;  required  the  orbit. 

Denote  by  k^  the  reciprocal  attraction  of  one  unit  of  mass  upon  an- 
other at  the  unit's  distance ;    then  will 

(3/J/,)  =  3/.J/,.^; 

and,  Equation  (27l), 

F=Jc^.  {M  -f  M)  .  u^  =  k^ .  m  .  u' ; 
in  which 

m  —  M  +  M^ (2'73)' 


200  ELEMENTS     OE    ANALYTICAL    MECHANICS. 

and.  Equation  (272), 

cV  It  hi .  m 

'^  *  ^  •  .  ^  -  C 

multiplying  by  2  cZ  w  and  integrating,     ^ 
whence 


d  a  =: 


^  /  ^       2l\.m  „ 


4  c 
tlie  negative  sign  being  taken,  because 


(') 


du  ^      [r/  ^  _  _dr_ 

da  da  r^  d  u. 


Place  under  the  radical  (  — r-  I  —  I  — — r  I  ■<  f^d  we  may  write, 
. . ■ —  ,,  \  4  f^  /        \  4  c'  /  •' 

t^^^  ,  1  —dn 

and  integrating, 


^'  .  m 

u — r- 

4  c 

a  +  9  =  cos 


^m 


+0 


in  which  9  is  the  constant  of  integration. 

Eeplacing  u  by  its  value,  taking  cosine  of  both  members  and  solving 
with  respect  to  r,  there  will  result 

k, .  in 


which  is  the  equation  of  a  conic  section,  having  its  pole  at  the  central 


MECHANICS    OF    SOLIDS.  201 

body.     To  find  the  precise  curve,  we  must  find  C.     To  do  this,  denote 

by  r^  the  initial  value  ot"  the  radius  vectoi',  and  by  e^  the  angle  whicli 

the  orbit  makes  with  r^  at  the  point  of  intersection  therewith.      Then,    /      _  , 

Equation  (275), 

Uv*-g  -^-J \  do.  r^tane/  '*"'* 


and  this  in  Equation  (274)  gives 


^^* 

'T-V..-^ 

I     ^  'A^ 

''■V: 

r 

J. 

^  /i;^  .  7/4 

ML' 

/  sin^ 

^/ 

4rr,   ' 

but. 

Equation 

(2G7), 

y 

'  -z^ 

.2  i 

A 

'.  ^'Ju.i,, 

1 

f; 

_    F/,- 

?•/  sm''  e,        4  c-       4  c- .  r,  ... 

in  which  F,  is  the  velocity  corresponding  to  1\ ;    hence,  ^  ^  j, 

F/  .  r^  —  2  k^  .  m  '       ^ 

^-'  477^  ' 

which,  substituted  in  the  equation  of  the  curve,  gives 

Ac" 


/           4  c**        /^^,       2k,.in\ 
^+V1+,TT:^^-(^/ ;— ).cos(a  +  ,) 


(276) 


and  comparing  this  with  the  general  polar  equation  of  a  coni;  section 
referred  to  the  focus  as  a  pole,  viz. : 

«.  (1  -  e') 

T  =  —^ 

1  +  e  cos  (a  +  (p)' 
we  find 

<-'(i-'')  =  *^. ("') 

^=^  +  X7:rf(''' — —)    •  •  •  •  <-'«) 

and  this  last  value  will  be  greater  or  less  than  unity,  according  as  F,' 

2  k^  .  m 
IS  greater  or  less  than  — . 

Multiplying  and  dividing  the  last  factor  by  M;  r,,  and  replacing  m 
by  its  value,  the  orbit  will  be  an  ellipse,  parabola,  or  hyperbola,  ac- 
cordino  as 


202  ELE.MENTS     OF    ANALYTICAL    MECHANICS. 

M.  .  v:-  >  -^l^'^IO  .  M. . ,.,.      ^"^"^^ 

That  is,  according  as  the  living  force,  of  the  primary  at  any  point  of 
its  orbit  is  less  than,  equal  to,  or  greater  than  twice  the  work  its  rela- 
tive weight,  at  that  point,  would  perform  over  a  distance  equal  to  its 
radius  vector.  So  that  a  primary  may  describe  any  of  the  conic  sec- 
tions as  well  as  the  ellipse,  the  only  condition  for  this  purpose  being 
an  adequate  value  for  its  velocity. 

Substituting  the  value  of  i^  in  E(|uation  (-77),  we  find 

k ,  .  m  .  r ,  ,   ^   s 

....     (279) 


2  k,  .  //*  -  V .  r,  ' 


and  denoting  the  semi-parameter  by  p,  the  equation  of  the  curve  gives, 
by  making  a  +  9  =  90°, 

4  r         V^ .  siu^  £  ,  r/ 
k^  .  rii  k^  .  Ill 

and  denoting  the  semi-conjugate  axis  by  i^, 


b,  =  ^ci  .  p  -  F  .  sin  e  .  r  \/^^^      .     .     .     (279)' 
'  ^    k^ .  m  ^       ' 

Whence  it  appears  that  the  nature-  of  the  orbit  and  its  transverse  axis 
are  independent  of  the  direction  of  the  primary's  motion,  Avhile  the 
conjugate  axis  is  dependent  upon  this  element. 

§  201. — The  consequence  of  Kepler's  third  law  is  not  less  important. 
Denote  tlie  periodic  time  of  the  primary  by  T^ ;    then,  Equation  (264), 

^'~        c       ' 

and  substituting  the  values  of  6„  ???,  and  c,  Equations  (279)',  (273)', 
and  (275)', 


MECHANICS     OF    SOLIDS.  ^OB 

and  for  another  body  wliose  mass  is  M,,,  about  tlie  same  central  body,_ 


T.,  =  2  7r«. 


1 


aud  by  division, 


(280) 


If  the  difference  of  the  masses  i/^  and  M,,  be  so  small  in  comparison 
with  M  as  to  make  its  omission  insensible  to  ordinary  observation, 
v.hich  is  the  case  in  the  solar  system,  the  above  may  be  written, 

Bat  by  Kepler's  third  law, 

whence 

'i'liat  is,  the  central  body  M  v.'ould  act  equally  on  the  unit  of  mass  of 
(;;ich  of  tlie  primaries  M,  and  M,,^  were  they  at  the  same  distance ;  so 
'Jiat  not  only  is  the  law  of  the  central  force  the  same,  but  the  abso- 
lute force  at  the  same  distance  is  the  same,  and  it  is  one  and  the 
same  force  that  keeps  the  planets  in  their  orbits  about  the  sun. 

§  202. — The  observations  of  Dr.  Maskelyne  on  the  fixed  stars,  show 
that  a  neighboring  mountain,  Schehallien,  drew  the  plumb-line  of  his 
instrument  sensibly  from  the  vertical  ;  and  those  of  Cavendish  and 
Baily  upon  leaden  and  other  balls,  demonstrate  this  power  of  attrac- 
tion to  reside  in  every  particle  of  matter  wherever  found  ;  and  that  it 
is  exerted  under  all  circumstances,  without  tlie  possibility  of  being  inter- 
cepted. It  is,  therefore,  concluded  that  matter  is  endowed  with  a  gen- 
eral gravitating  principle  by  which  every  particle  attracts  every  other 
particle,  and  according  to  the  law  before  given. 

PERTURBATIONS. 

§  203. — Granting,  for  the  present,  that  univei'sal  gravitation  is  a 
principle  of  nature,  and  denoting  the  distances  of  the  several   bodies  of 


20-i:  ELEMENTS     OF     ANALYTICAL    MECHANICS. 

the  system  from  tlie  central  by  r  with  subscript  accents  corresponding 
to  those  of  the  bodies  to  Nvhich  they  belong,  and  employing  the  same 
notation  in  regard  to  the  co-ordinates,  we  shall  have 

M  M     r'  v' 

[MM),  =  k  .  — — '  .  -  =  /.• .  M.M^  .  -3 ; 

M,,    x"       ,     _      M.x" 


r  -^     r  '       '         r  '     ' 


•Z  (i/,  J/,  Jx  =  X-  .  V 


2  (if  1/,J.  =-  1<  M.^—^;-—  =  k  .  i/.  2 


{x"  -x'  f-]-{y"  -yf+{z'  -z'Y     v/^a;"-a;7-|-(2/"-y'f+(.:;"-^'f  ' 


which,  substituted  in  first  of  Equations   ("258),  give 


d^x        ,r,,r,    ,rs     x'         il/,.r"   ,     1  J\£,.  M,,.{x"-x')  T 


but 

x"—x'  „         1 


d 


l{x'-  x'f-\-  {y"-y'f+  {z''-zJY       dx  v/(*"-a;'f-f  (?/"-y')=  +  (^''-«')= ' 

and  making 

the  last  term  of  the  equation  above  becomes 

1      dX 

and 


'i^_.[(..+  .,).l,_.^4^'  +  ^^.-]  =  „. 


Make 


i^— 1 T h«-v;c. —    (282) 


then  will 


dR_M,,x"      M„,x"'  dX      _  X"  dX 

dx'-     rj     +      r,,/""^  M,.dx'~^'^^"'V;}~M;jU''' 

which,  substituted  above,  give,  after  treating  the  other  two  of  Equations 
(2rj8)  in  the  same  way, 


MECHANICS     OF    SOLIDS.  205 


The  curve  whicli  would  .be  described  by  the  primary  about  the  central, 
under  the  reciprocal  action  of  these  two  bodies  alone,  and  which  avc 
have  seen  is  a  conic  section,  is  called  the  undiaturhed  orbit  of  the  pri- 
niary.  That  which  it  actually  describes  under  the  joint  action  of  all 
the  bodies  of  the  system,  is  called  the  disturbed  orbit.  The  undisturbed 
orbit  is  given  by  the  first  two  terms  of  Equations  (283) ;  the  disturbed 
by  all  three.  The  departures  of  the  disturbed  from  the  undisturbed 
orbit  are  called  ^^er<t<rio</oHS,  and  the  last  terms  of  Equations  (283), 
which  determine  them,  are  called  2)eriti7'bali)if/  functions.  The  construc- 
tions of  the  perturbating  functions  are  given  in  Equations  (281)  and 
(282),  and  the  methods  of  computing  their  values  are  greatly  facihtated 
by  the  principle  of  the 

COEXISTENCE    AND    SUPERPOSITION    OF    SMALL    MOTIONS. 

§  204. — Denote  by  d^j,  0^^^,  etc.,  numerical  quantities  which  depend 
upon  the  perturbating  actions  of  the  bodies  whose  masses  are  M^^,  -^^t,,i 
ikc,  and  of  which  the  values  are  so  small  as  to  justify  the  omission  of 
all  terms  into  which  their  products  enter  as  factors,  in  comparison  with 
such  as  contain  them  singly.  The  co-ordinates  of  31^,  at  the  time  t, 
when  undisturbed,  being  x'  y'  z\  become,  when  the  body  M^  is  disturbed 
by  M^^  at  the  same  time, 

and  for  the  same  reason,  when  also  disturbed  by  J/^^^, 

^  +  9,;^;'  -f  e„,  (a;'  -1-  Q,,x') ;    y'  -f  Q  j)'  -{-  0,„  [xj  -f-  0,,;y') ;    2'  -f  Q„z  -f  0,„(«'  +  Q.?') , 
or,   performing    the    multiplication    and    omitting    the    terms    containing 

•''■'  +  .f'(0,-fO^,,);    y' +  y' (0,,  +  0,,,) ;    ^'-f 2:' (0,4-0, J; 


i 


206  ELEMENTS    OF    ANALYTICAL    MECHANICS, 

in  the  same  way,  when  also  disturbed  by  M^^^^^ 

:r'+.r'(o,,+0,.,+0.,J;  !/'+y'(^.+0,.+0>.J;  ^'+^'(^.+9,,,+o,„,); 

and  for  the  simultaneous  disturbance  of  all  the  bodies  of  the  system, 

in   which   .r'.2  0^^,  y',2  0^^,  z'.^Q^^  are   the  .increments  oi  x  y' z'  re- 
spectively, due  to  the  joint  action  of  all  the  disturbing  bodies.     Now  let 

z(  =  9  {x'  y'  2'), 

in  which  9  denotes  any  function  of  x'  y'  z'.     Differentiating,  we  have 

du 


d  u       ,  ^  .         du        ,^ 
d  x'  d  y 


+ 


dz' 


and  j^erforming  the  multiplications  indicated,  avc  have 


du       ,  .  d  u        ,  ^         d  II       ,  ^ 

dx  dy  dz 

du       ,  „  d  11        ,  ^  d  n       ,  ^ 


d  u 


d  u 
a  X 


1+        «kc 


+ 


etc.        + 

etc.  + 


Whence  it  appears  that  the  perturbation  in  u  or  9  {x'  ?/'  z'\  is  equal  to 
the  sum  of  the  separate  perturbations  due  to  each  of  the  perturbating 
bodies,  supposing  the  others  not  to  exist.  The  practical  effect  of  this 
principle  is  to  reduce  the  problem'  of  the  perturbations  from  one  of 
several  to  one  of  a  single  perturbating  body,  and  to  give  rise  to  what 
is  known  as  the  problem  of  the  three  bodies,  viz. :  the  central,  primary, 
and  perturbating. 


UNIVERSAL     GRAVITATION. 


§  205. — From  all  of  which  it  is  manifest  that  cither  Kepler's  laws 
can.iiot  be  rigorously  true,  or  universal  gravitation  is  not  a  Princi[)le  of 
Xature.      Now,  in  pt^int  of  fact,  observations  of  far  greater  nicety   than 


MECHANICS     OF    SOLIDS,  207 

those  of  Kepler  prove  that  bis  laws  are  not  accurately  true,  though 
they  differ  but  slightly  from  the  truth  ;  a  circumstance  arising  entirely 
from  the  fact  of  the  great  mass  of  the  sun  as  comj^ared  with  the  sum 
of  the  masses  of  all  the  planets.  Were  there  but  a  single  body  in  ex- 
istence besides  the  sun,  it  would  describe  accurately  an  elliptical,  para- 
bolic, or  hyperbolic  orbit  about  the  centre  of  the  sun,  depending  upon 
its  living  force  and  the  sun's  attraction.  A  third  body  would  derange 
this  motion  and  cause  a  departure  from  this  simple  path,  and  the  de- 
gree of  the  disturbance  would  depend  upon  the  mass,  distance,  and  di- 
rection of  the  disturbing  body  as  compared  with  those  of  the  sun.  The 
same  remark  would  apply  to  a  fourth,  fifth,  and  to  any  number  of  addi- 
tional bodies.  The  disturbed  orbits  in  the  solar  system  have  been  com- 
puted by  Equations  (283),  and  the  complete  harmony  which  is  found 
to  subsist  between  the  numerical  results  deduced  from  theory  and  ob- 
servation, is  the  strongest  possible  evidence  in  support  of  the  Law  of 
Universal  Gravitation. 

If  the  principal  plane  of  the  solar  system,  as  determined  at  different 
and  remote  periods,  be  found  to  have  undergone  no  change,  this  will 
show  that  the  system  is  uninfluenced  by  the  action  of  the  fixed  stars 
and  other  distant  bodies,  and  its  centre  of  inertia  will,  §  198,  either  be 
at  rest  or  be  moving  uniformly  through  space  in  a  right  line ;  but  if 
the  principal  plane  be  found  to  have  changed  its  place,  it  will  be  a  sign 
that  the  system  is  in  motion,  and  that  its  centre  of  inertia  is  describing 
a  curvilinear  patli  about  some  distant  centre. 

§  206. — Thus  much  for  the  larger  bodies  of  nature.  But  these  are 
themselves  built  up  of  innumerable  molecules  whicli  are  ever  on  the 
move  about  their  respective  places  of  relative  rest.  The  molecular 
forces  within  the  range  of  their  natural  action  vary  directly  as  the  dis- 
tance from  their  respective  centres.  Let  it  be  required  to  determine 
the  nature  of  the  orbits  under  this  law.     Then  will 

F  =  m  h, .  r  = ; 

u 

which,  in  Equation  (272),  gives 

cV'  u  k. .  in 

da'  4  c-  %•* 


20S  ELEMENTS     OF    ANALYTICAL    MECHANICS. 


multiplying  hj  2  d  u,  and  integrating,  we  find 

d  u^         „       ^      k 
da  4  c-  u 


d  II?         „       ^      k, . «;  .      , . 

+  "  =^-7^ (284) 


from  wliich  we  get 

%i  d  u 


d  a 


Ac' 
the  negative  sio-n  beino;  taken,  because 

O  O  O  ' 


beca 


k, .  m 


d  , 
dit  \  rf  d  r 

da         da  f''  d  a 

Placing  -J  C^  —  i  (7^  under  the  radical,  we  may  write 

1  —  2  zf  d  u 


{2uy 


4c^       /  C-      k,.m 


^«  =  i 


4c^ 
and  integration, 


2  (ot  +  f)  =  cos 


—  1  2 


4c^ 


in  which  (p  is  the  constant  of  integrating. 

Taking  cosine  of  both  members,  replacing  u  by  its  value  and  solving 
with  respect  to  ?•,  we  find 

1 

r  =      , 

V  ^C+  }s\/  C  -^^  .cos2{a  +  cp) 

Denote  by  7\  the  radius  vector  which  is  normal   to   the  orbit ;   corre* 
sponding  to  this  value  we  have 


d  u 


and,  by  Equation  (284), 


1     ,   k,.m.r,\ 
^  -  ,"^  +       4  c'       ♦ 


MECHANICS    OF    SOLIDS.  209 

and  because 

cos  2  (a  +  y))  =  cos^  (a  +  (p)  —  sin'  (a  +  (p), 

tlie  above  reduces  to 

1 


/  1        „  ,       k,  .ni .  r/'    ...      ,      , 

V  -2  ^os'  («  +  <?)+  ■ — j-i —  sm=  (a  +  9) 


(285) 


which  is  the  equation  of  an  ellipse  referred  to  its  centre  as  a  pole,  the 
semi-axes  being 


r.  and   —  y   - — 


k, .  m 

§  2C7. — The  time  required  to  describe  the  entire  ellipse  being  deno- 
ted by  T,  we  have,  Equation  (2G4), 

r^  k.  .m  if     i- 

T  = '- —  =  2  Try ; 

?V  .  c  ^i-  "^ 

and  replacing  m  by  its  value,  Equation  (293)', 


T=27tV- ^ — ^- (286) 

Thus  the  time  is  wholly  independent  of  the  dimensions  of  the  orbit, 
and  will  be  the  same  in  all  orbits,  great  and  small.  This  result  finds 
its  application  in  the  subject  of  acoustics,  thermotics,  optics,  &c. 

§  208. — Let  us  conclude  the  planetary  motions  with  the  centrifugal 
force  on  its  surface,  arising  from  the  rotation  of  one  of  these  bodies, 
say  the  earth,  about  its  axis. 

If  F,  denote  the  angular  velocity  of  a  body  about  a  centre,  then  will 
F— pF|,  and  Equation  (270)  becomes 

The  earth  revolves  about  its  axis  A  A'  once  in  twenj;y-four  hours, 
and    the    circumferences    of    the    parallels    of    latitude    have    velocities 

14 


210 


ELE^[E^'TS     OF     ANALYTICAL     MECHANICS. 


which  diminish  from  the  eq  lator  to 
the  poles.     The  law  of  this  diminu- 
tion,   on    the    supposition    that    the    h4 
planet  is  a  sphere,  is  given  by 

in  which  31  is  the  bod\''s  mass,  F, 
the  earth's  angular  velocity,  and  H' 
the  radius  of  one  of  its  parallels  of 
latitude. 

Denoting    the    equatorial    radius   C  U  =  C  F,  by  M,  and    the    angle 
C  P  C  =  P  C  jE,  which  is  the  latitude  of  the  place,  by  9,  we  have 

R'  ^=  R  cos  <p  ; 

which  substituted  for  R'  above,  gives 

F^  =  MV^^Rco^^ (286)' 


The  only  variable  quantity  in  this  expression,  when  the  same  mass 
is  taken  from  one  latitude  to  another,  is  9;  whence  wc  conclude  that 
the  centrifugal  force  varies  as  the  cosine  of  the  latitude. 

The  centrifugal  force  is  exerted  in  the  direction  of  the  radius  R'  of 
the  parallel  of  latitude,  and  therefore  *in  a  direction  oblique  to  the  ho- 
rizon  T  T' .     The  normal  and  tangential  components  are,  respectively, 

F,  .  cos  9  :=z  M  V,'  R  cos^  9, 
F^ .  sin  9  r=  M  V{^  R  .  sin  9  cos  9  =:  i  i)/  V^^  i2  sin  2  9  ; 

whence  we  conclude,  that  the  diminution  of  the  iveights  of  bodies 
arising  from  the  centrifugal  force  at  the  earth's  surface,  varies  as  the 
square  of  the  cosine  of  the  latitude;  and  that  all  bodies  are,  in  con- 
sequence of  the  centrifugal  force,  urged  toivards  the  equator  by  a  force 
lohich  varies  as  the  sine  of  tivice  the  latitude. 

At  the  equator  the  diminution  of  the  force  of  gravity  is  a  max- 
imum, and  equal  to  the  entire  centrifugal  force;  at  the  poles  it  is  zero. 
The  earth  is  not  perfectly  spherical,  and  all  observations  agree  in  de- 
monstrating that  it  is  protuberant  at  the  equator  and  flattened  at  the 
poles,  the  difference  between  the  equatorial  and  polar  diameters  being 
about  twenty-six  English   miles.     If  we  suppose  the  earth  to  have  been 


irli 


MECHANICS     OF     S0L1L.S.  211 

at  one  time  in  a  state  of  fluidity,  or  even  approaching  to  it,  its  present 
fio-ure  is  readily  accounted  for  by  the  foregoing  considerations. 

To  find  the  value  of  the  centrifugal  force  at  the  equator,  make,  in 
Equation  (286)',  M=:  1  and  cos  9  =  1,  which  is  equivalent  to  suppo- 
sing a  unit  of  mass  on  the  equator,  and  "vve  have 

in  which,  if  the  known  radius  of  the   equator  and   angular  velocity  be 
substituted,  we  shall  find 

F^=  F,\  J?  =  0,  1112. 

To  find  the  angular  velocity  with  which  the  earth  should  rotate,  to 

make    the    centrifugal    force    of   a    body    at    the    equator    equal    to    its 

weight,  make 

/ 
y  =  32,  1937  =  V/'E; 

/  '  . 

in  which  32,  1937  is  the  force  of  gravity  at  the  equator. 

Dividing  the  second  by  the  first,  we  find 

32,1937        V/'       ^^^ 

— z=-^  =  289,  nearly ; 

0,1112        F,'  '  •^  ' 

Avhence, 

that  is  to  say,  if  the   earth  were  to  revolve  seventeen  times  as  fast  as 
it  does,  bodies  would  possess  no  weight  at  the  equator. 

IMPACT    OF    BODIES. 

§  209. — When  a  body  in  motion  comes  into  collision  with  another, 
either  at  rest  or  in  motion,  an  iinparA  is  said  to  arise. 

The  action  and  reaction  which  take  place  between  two  bodies,  when 
pressed  together,  are  exerted  along  the  same  right  line,  perpendicular  to 
the  surfaces  of  both,  at  their  common  point  of  contact.  This  arises  from 
the  symmetrical  disposition  of  the  molecular  springs  about  this  line. 

When  the  motions  of  the  centres  of  inertia  of  the  two  bodies  are 
2Mrallcl  to  this  normal  before  collision,  the  impact  is  said  to  be  direct. 

When  this  normal  passes    through    the    centres    of  inertia    of  both 


212 


ELEMENTS     OF    ANALYTICAL    MECHANICS. 


:^ 


bodies,    and   the   motions    of   these    centres   are    along    that   line,    the 

impact     is    said    to     be    direct    and 

central. 

When  the  motion  of  the  centre 
of  inertia  of  one  of  the  bodies  is 
along  the  common  normal,  and  the 
normal  does  not  pass  through  the 
centre  of  inertia  of  the  other,  the 
impact  is  said  to  be  direct  and 
eccentric. 

When  the  path  described  by  the 
centre  of  inertia  of  one  of  the  bodies, 
makes  an  angle  with  this  normal, 
the  impact  is  said  to  be  oblique. 

When  two  bodies  come  into  col- 
lision, each  will  experience  a  pres- 
sure from  the  reaction  of  the  other ;  and  as  all  bodies  are  more  or 
less  compressible,  this  pressure  will  produce  a  change  in  the  figure 
of  both  ;  the  change  of  figure  will  increase  till  the  instant  the  bodies 
cease  to  appa'oach  each  other,  when  it  will  liave  attained  its  maximum. 
The  molecular  spring  of  each  will  now  act  to  restore  the  former 
figures,  the    bodies  will  repel  each    other,  and  finally  separate. 

Three  periods  must,  therefore,  be  distinguished,  viz.  :  1  st..  that 
occupied  by  the  process  of  compression  ;  2d.,  that  during  which  the 
greatest  compression  exists ;  3d.,  that  occupied  by  the  process,  as 
far  as  it  extends,  of  restoting  the  figures.  The  force  of  restitution 
must  also  be  distinguished  from  tlie  force  of  distortion;  the  latter 
denoting  the  reciprucal  action  exerted  between  the  bodies  in  the 
first,  and  the  former  in   the   third    period. 

The  greater  or  less  capacity  of  the  molecular  springs  of  a  body 
to  restore  to  it  the  figure  of  which  it  has  been  deprived  by  the 
application  of  some  extraneous  force  when  the  latter  ceases  to  act, 
is  called    its  elasticity. 

The  latio  of  the  force  of  restitution  to  that  of  distortion,  is  the 
measure  of  a  body's  elasticity.  This  ratio  is  sometimes  called  the 
co-efficient  of  eiaslicit//.      When    these    two    forces   are   equal,  the    ratio 


MECHANICS     OF     SOLTDS. 


213 


is  unity,  and  the  body  is  said  to  be  'perjlctlij  eluslic ;  wlieu  the 
liitio  is  zero,  the  body  is  said  to  be  non-clustic.  There  arc  no  bodies 
that  satisfy  these  extreme  conditions,  all  being  more  or  less  elastic 
but   none  perfectly  so. 

Let  the  two  bodies  AB  and  A'  B\  the  former  moving  along  the 
line  H T,  and  the  latter  along 
//'  T',  come  into  collision  at  the 
point  0.  Through  0,  draw 
the  common  normal  K L.  De- 
note the  angle  H  O  N  by  (p, 
and  H'  EX  by  9' — these  being 
llic  angles  which  the  directions 
of  the  two  motions  make  v.ith 
the  normal.  Also  denote  the 
velocity  and  mass  of  the  body 
AB  by  V  and  M  respectively,  and  the  velocity  and  mass  of  A'  B' 
by    V  and  M'. 

The  components  of  the  quantity  of  motion  of  the  two  bodies  in 
the  direction  of  the  normal  and  of  the  perpendicular  to  the  normal, 
will   be 

M  V  cos  9,     AB  V  cos  9'     and     M  V  sin  9,     M'  V  sin  9'. 

The  former  of  these  components  will  alone  be  involved  in  the 
impact ;  for  if  the  bodies  were  only  animated  by  the  latter,  they 
would  not  collide,  but  would  simply  move  the  one  by  the  other. 
For  simplicity,  let  the  body  AB  be  spherical;  the  normal  will 
pass    through   Its   centre  of  inertia. 

Denote  by  ti,  the  velocity  of  the  body  ^  ^  in  the  direction  of 
the  normal  at  the  instant  of  greatest  compression,  and  by  u'  the 
velocity  of  the  body  A'  B'  at  the  same  instant  in  the  same  direction. 
Then    will 


V  cos  9  —  M,     and      V  cos  9'  —  u' 


(287) 


be  the  velocities  lost  and  gained  in  the  direction  of  the  normal,  and 
i/(Fcos9  -  li),     and     i1/'(7'cos9'  -  u')    ■  -  •  (288) 


214  ELEMENTS     OF     ANALYTICAL     MECHANICS. 

be  the  forces  lost  and  gained  at  the  instant  of  greatest  compression  ; 
and   hence, 

i/(  F  cos  9  -  u)  J-  M'  ( V  cos  9'  -  u')  =  0  ;  •     •     (289) 

and  denoting  the  angular  velocity  of  the  body  A' B'  by  V/ ,  the 
distance  G'  D  from  the  centre  of  inertia  of  A'  B'  to  the  normal 
by  e,  and  the  principal  radius  of  gyration  of  A'  B',  with  reference 
to    the    instantaneous  axis  by  k^ ,  then  will 

_  3/(Fcos9  -u).e  ^ 

'  -         M'k;^  ^^  ^ 

and  since  the  velocity  m  must  be  equal  to  that  of  the  point  D  at 
the   end   of  the   lever   arm  e,  we   have 

u  =  zc'-\-e.V/ (291) 

Substituting  the  values  of  ti  and  ti'  from  this  equation  successively 
in   Equation  (289),  we  find 

M  F  cos  ^  +  M'  V  cos  9'  +  M'e  F/ 


M  +  M' 

M  F  cos  9  +  M'  V  cos  9'  -  Me  V/ 
M  +  M' 


(293) 


After  the  instant  of  greatest  compression,  the  molecular  springs 
of  the  bodies  will  be  exerted  to  restore  the  original  figures,  and 
if  c  denote  the  co-efficient  of  elasticity,  then  will  the  velocities  lost 
hy  AB  and  gained  by  A'  B'  during  the  process  of  restitution  be, 
respectively, 

c{^V  cos  9  —  t()     and     c  {V  cos  9'  —  u') ; 

and    the    entire   loss    of  yli>,  and  gain  oi  A'  B',  will  be,  respectively, 

V  cos  9  —  z<  4-  c  (  F  cos  9  —  «),    and     V  cos  9'  —a'  -\-  c{V'  cos  9'  —  u'). 

Also    the    gain    of    angular  velocity    of  the    body  A'  B\  during    th« 
process  of  restitution,  will  be 

,  (  Fcos  9  —  m)  ,  e  ^[ 


MECHANICS     OF    SOLIDS.  215 

and  the  whole  angular  velocity  produced  by  the  impact  and  denoted 
by    Vj,  will    be   given   by    the    equation, 

TT         /n     ,     V  (Fcosffl  — «)  e   M 

r,  =  (l+c)i ^-^-Jf.      ■     ■     ■      (294) 

Denoting   the  velocities  of  A  B  and    A'  B',  after   the   collision    by 

V  and  v',  and  the  angles  which  the  directions  of  these  velocities 
make  with  the  normal  by  6  and  6',  respectively,  then  will 

V  cos  t)  =:  Fcos(p  —  Fcos(p  +  ti—c  (Fcos(p—  u)  —  [l  -\-  c)  u  —  c  Fcos cp, 

i;'cos  &'=  F'cos  9'—  F'cos  (p'+«'— c(  F' coscp'—u')=z[l-]-c)u'—cV' cos  9', 

and  replacing  the  values  of  u  and  «',  as  given  by  Equations  (292) 
and  (293), 

,  31  V cos cp+M'  V'cos(p'+M'e  V' 
vcosdz=(l-^c)  ^j.-^  ^^, .-cFcos9,   (295) 

„     ,,       ,iJ/Fcos(p+7l/'  F'cos(p'-.¥e  P7  ,„  ,,^^ 

?;'cosr  =  (l+c)  ^-^-- -Tjj- -c  F'cos9'(29G) 

Moreover,  because  the  effects  of  the  impact  arising  from  the  compo- 
nents of  the  quantities  of  motion  in  the  direction  of  the  normal  will 
be  wholly  in  that  direction,  the  components  of  the  quantities  of 
motion  before  and  after  the  impact  at  right  angles  to  the  normal  will 
be  the  same,  and  hence 

V  sin  ^  r=   Fsin  9, (297) 

v'  sin  ^'  =  V  sin  9' (298) 

Squaring  Equations  (295)  and  (297)  and  adding;  also  Equations 
(296)  and  (298)  and  adding,  we  find  after  taking  square  root,  and 
reducing  by  the  relations 

cos3  ^  +  sin2  ^  =  1  ;    cos3  (3'  +  sin^  ^'  =  1  ; 

r,         ,i/Fcos9  +  i/'  V'qosq?'-{-M'c  F/       Z  ^71  ■  1      /ono\ 

v=^\{\  +  c) if  +  M'      ^-cFcos9]2+  F2sm29.  (299) 


1^,          ,Jf  Fco.s9  +  il/'F'cos9'~ilft.'F/  ,    .        ,  /onn\ 

v'-\/\{\+c)-    J/   I    ,)/r -'-cF'cos9']-+  F2sm29'-(30{)) 


216         ELEMENTS     OF     ANALYTICAL     MECHANICS. 

Dividing  Equation  (297)    by  Ecjuation  (295).  and    Equation  (298)  by 
Equation  (290),  we  have, 

V .  sin  (p 
*^^^^  ^  7         ,  M  7cos  CD  +  M'  V  cos  m'  +  M' e  V!         ~'  '^^^^^ 

(1  +  ^) jf:^^' "  ^^°^^ 

V .  sincp' 

tan  6'  — :rry^ ■ — ,,.  „. -, T7 — 777 (302) 

,1/  Fcos©  +  M'  ¥■  cosm'  —  Me  V.'         ^^,  ,^       ' 

(1  +  C) ■ ^rr~, TTT- cT^'coso)' 

Equations  (290)  and  (292),  will  give  the  values  of  n  and  F/,  in 
known  terms,  and  these  in  Equations  (294),  (295)  and  (296)  "will 
give  the  values  of  F^,  v,  and  v\  and  all  the  cii'cunistances  of  the 
collision  will  be  known. 

g  210. — If  the  bodies  be  both  spherical,  then  will  e  =  0,  and  Equa- 
tion (294)  gives  F,  =  0 ;  and  Equations  (299)  and  (300),  (301)  and 
(302),  become 

v=^m+c)^^'^^^^^^^f^^^^-oY^.,f+V^.^^,  •  •  •  (303) 

1/   sill  GD 

tan^  =  jVTT ; — T77T77 ~r '    '   *   (305) 

,  J/Fcos  (D  +  M'  V  cos  o'         ^^  ^       ' 

(1  +  .) j^-^-j^, c  F  cos  9 

V  sin  (p'  /oA->\ 

tan «'  = -jTrcosf  +  J/'  r^3J? r ,  •  •  f^*''' 

(1  +  '^) ihtm- — -  - '  ^  ""  " 

The  Equations  (303)  and  (304)  will  make  known  the  velocities, 
and  (305)  and  (oOG)  the  directions  in  which  the  bodies  will  move, 
after  the   impact. 

Now,  suppose  the  body  A!  B'  at  rest,  and  its  mass  so  great  that 

the  mass   of    A  B    is    insignificant    in    comparison,    then    Avill    V   be 

M 
zero,  M'  may  be  written  for  M  +  J/'  and   — —  ^\•ill  be  a  fraction  so 


MECHANICS     OF     SOLIDS. 


217 


small  that   all    the   terms    into    which   it    enters  as    a   factor    may    Lt 
neglected,  and  Equation  (303)    becomes 


V  =1  V  -y/  c-  COS"  9  +  sin'-^  ^  ; 


and   Equation   (305), 


tan  & 


tan  cp 


(307) 


The  tangent  of  ^  being  negative,  shows  that  the  angle  xYT/A", 
which  the  direction  of  A  B^s  motion 
makes  with  the  normal  iViV  after  the 
impact,  is  greater  than  90  degrees ;  in 
other  words,  that  the  body  A  B  is 
driven  back  or  reflected  from  A'  B'. 
This  explains  why  it  is  that  a  cannon- 
ball,  stone,  or  other  body  thrown  ob- 
liquely against  the  surfixce  of  the  earth, 
will  rebound  several  times  before  it 
comes   to   rest. 

If  the  bodies  be  non-eUxstic,  or,  which  is  the  same  thing,  if  c  be 
zero,  the  tangent  of  ^  becomes  infinite ;  that  is  to  say,  the  body 
A  B  will  move  along  the  tangent  plane,  or  if  the  body  A'  B'  were 
reduced  at  the  place  of  impact  to  a  smooth  plane,  the  body  A  B 
would   move   along   this   plane. 

If  the  body  were  perfectly  elastic,  or  if  c  were  equal  to  unity, 
which    expresses   this   condition,  then    would    Ec[uation  (307)  become 


tan  1^ 


tan  (p 


(308) 


which  means  that  the  angle  N  HF  =  E H N'  becomes  equal  to 
KHN'.  The  angle  EHX'  is  called  the  angle  of  incidence,  the 
angle  KHN',  commonly,  the  angle  of  reflection.  Whence  we  see, 
that  when  a  perfectly  elastic  body  is  thrown  against  a  smooth,  hard, 
and  fixed  plane,  the  angle  of  incidence  will  be  equal  to  the  angle 
of  reflection 

If  the  angles  9  and  9'  be    zero,  then  will    cos  9^1,    cos  9'  —   ], 


218  ELEMENTS     OF    ANALYTICAL    MECHANICS. 

sill  t?  =  0,    sin  (p'  =  0 ;   the    impact   will   be   direct   and   central,  and 
Equations  (303)  and  (304)  become 

,  J/  F  +  Jf  '  V 
V  =(l  +  c) ,^  ,     ,„ cV, 

^'  =  i'^^)       M+M'      --^- 

and  passing  to  the  limits,  non-elasticity  on   the   one    hand  and  perfect 
elasticity  on  the  other,  we  have  in  the  first  case,  c  =  0,  and 

MV  +  M  V 

'  =  ^^r^ir- (^^^) 

'  =  -jrr^r- (^^^^ 

and  in   the   second,  c  =  1,  consequently, 

» =--  ""-mTm^  -' <«") 

3fV  4-  M'  V 


.    V/  CONSTEAINED   MOTION. 

§211. — Thus  far  we  have  only  discussed  the  subject  oi  free  motion. 
We  now  come  to  cnnstraincd  motion. 

Motion  is  said  to  be  constrained  when  by  the  interposition  of 
some  rigid  surfiice  or  curve,  or  by  connection  with  some  one  or 
more  fixed  points,  a  body  is  compelled  to  pursue  a  path  different 
from    that   indicated   by    the   forces   which  impart   motion. 

§  212. — The  centre  of  inertia  of  a  body  may  be  made  to  con- 
tinue on  a  given  surface,  by  causing  it  to  slide  or  roll  upon  some 
other  rigid  surface. 

§213. — We  have  seen,  §128,  that  the  motion  ot  translation  of 
the   centre    of  inertia,    and    of  rotation    about    that   t)oint,  are    wholly 


MECHANICS     OF    SOLIDS. 


219 


independent  of  one  another,  and  the  generality  of  any  discussion 
relating  to  the  former  will  not,  therefore,  be  affected  by  making, 
in   Equation  (40), 

Sep  =  0;     3-^  =  0;     Szi  =  0; 
Yhich  Avill  reduce  that  equation  to 

iP  X 


(2  P  cos  a  —  "--^  ■  2  m)  S  x, 


+  (2  P  cos  ,S 


+  (2  P  cos  7 


J2y 
dp 


2  m)  5  y, 


2  m)  0  z, 


\  =  0. 


Making 

2  7?i  =  J/;     2Pcosazr:.Y;     2Pcos/3=F;     2Pcos7  =  Z; 
and  omitting  the  subscript  accents,  we  may  Avrite 

(x-i/.^)  A.  +  {r-M--^  iy  +  (Z-M.'lf)  fc=0.(313) 

Now,  assuming  the  movable  origin  at  the  centre  of  inertia,  and 
supposing  this  latter  point  constrained  to  move  on  the  surface  of 
u  hich  the  equation   is 

L  =  F{xyz)=^0, (314) 

the  virtual  velocity  must  lie  in  this  surface,  and  the  generality  of 
Equation  (313),  is  restricted  to  the  conditions  imposed  by  this  cir- 
cumstance. 

Supposing  the  variables  x  r/  z,  in  the  above  equations,  to  receive 
the  increments  or  decrements  S  x,  5?/,  o  z,  respectively,  we  have,  from 
the  principles  of  the  calculus. 


dL  dL  dL 

~l~-8x  +  ~-  -Sy  +  —--Sz  =  0. 

dx  d  y  dz 


(315) 


Multiplying  by  an  indeterminate   quantity  X,  and  adding   the   product 
to  Equation  (313),  there  will  result 


(x-i/.^  +  x   '^^ 


dp 
dl 

fZ2 


+  (^-^--^+-^)^>  h^' 


dy 


{ r,        .t     (P-2  dL\   ^ 


220 


ELEMENTS     OF    ANALYTICAL    MECHANICS. 


The  quantity  X,  being  entirely  arbitrary,  let  its  value  be  such  as  to 
reduce  the  coefficient  of  one  of  the  variables  S x,  8y,  8  z,  say  that  of 
ox,  to  zero;  and  there  will  result 


^-^•?l  +  --^  =  o. 


(816) 


and 


(r->/.l^+x4^)  S,j  +  (z-.V.'^4  +x.!i^)  S-.  =  0.  (317) 

V  dfi  d  yy      -^         \  d  i"  dz  /  ^        ■' 

Now  in  Equation  (315),  oy  and  J.r  may  be  assumed  arbitrarily,  and 
ox  will  result;  hence  <)y  and  <)  z  in  Equation  (317)  niay  be  regarded 
as  independent  of  each  other,  and  by  the  principle  of  indeterminate 
coefficients, 


d  t"  dy  I 


^-^■?I  +  -^  =  «.J 


and   eliminating  X  by  n:ieans  of  E(juation  (316),  we  find, 


(318) 


(  r  -  if .  ^ )  .  ^  -  (X  -  Jf .  ^ )  .  ^  =  0, 
\  dl-y       dx  \  df/      dy  ' 

\  dt-y       dy         \  dt-y 


■y 
djj_ 

liz 


=  0; 


119) 


which,  with    the  equation  of  the  surface,   will   determine  the  place  of 
tlie   centre  of  inertia  at  the  end  of  a  given  time. 


MOTION    ON    A   CUKVE   OF   DOUBLE    CUKVATUKE, 

§214. — If  the  centre  of  inertia  be  constrained  to  move  upon 
two  surfaces  at  the  same  time,  or,  which  is  the  same  thing,  upon 
a  curve  of  double   curvature  resulting  from  their  intersection,  take 


L  =  F{xyz)  =--  0,   ) 
H^F'{xyz)  =  0;  f 


(320) 


MECHANICS     OF     SOLIDS. 


221 


from  which,  by  the  process  of  differentiating  and  rejjlacing  o?.r,  dy,  dz. 
by  tlie    projections  of  the   virtual    velocity, 


d  L   ^  d  L  ^  d  L     ^ 

S,c  ^  Sy  +  -—.Sz  .-=  0; 
d X                 dy  dz 

dH  dH  dH 


(321) 


(322) 


Multiplying  the  first  of  these  by  X,  and  the  second  by  X',  adding  the 
products  to  Equation  (313),  and  collecting  the  coefficients  of  8  x^  8y, 
and  5  z,  we  have 


(x  -  M. 


d'^  X 


+  ^ 


+  ^ 


d  X  / 


+  {y 


dL 

d  X 

dL  ,    d  H\   . 

dy  dy/ 

dz  d  z  / 


=  0 


(323) 


No\y  the  coefficients  of  two  of  the  three  variables  ox,  8y  and  6  z, 
say  those  oi  S x  and  8y,  may  be  made  equal  to  zero  by  assigning 
proper  values  for  that  jiurpose  to  the  indeterminate  quantities  X  and 
X',  in  which  case,  since  8  z  is,  not  equal  to  zero,  its  coefficient  must 
also  be  equal  to  zero ;    whence 


, ,     d"^  x 
X  —  M  •  -^^+  X 


Y  -  M  ■ 

Z  -  M- 


df" 

d'^z 

df 


+  X 


dL 
d  x 

d  L 

dy 


4-X' 


d  L 

d  z 


dH 

dx 

dH 

dy 

dH 

d  z 


=  0, 

=  0, 
=  0. 


(324) 


and  eliminating  X  and  X',  there  will  result 
fp  .i'\      fd  L     d  H 


/  X.       ,  r    f^'  'A      /f^  L     dH         d  L     dH  \ 

iX  —  M-  -T^  )  •  (  -~~  •  -Y 7—  •  -J —  ) 

\  a  ^ /      V  a s       d  y  dy      d  z  / 

/^^       ,rf^''/\      f^' L     dH         dL     dH\ 
+  (Y-M--^)  .  (- J-) 

\  dl-y       \dx       dz  d  z       dx  y 

/.,        ,.  rf'2\     f<^  L     dH         dL     dH\ 
\  df/\ d  y       d X  d x       dy  / 


\  =.  0.   (325) 


222 


ELEMENTS  OF  ANALYTICAL  MECHANICS. 


which,  with  the    equations  of  the    surfaces,  is    sufficient    to    determine 
the  co-ordinates    of  the    centre   of  inertia  when  the   time   is   given. 


(326) 


§  215. — If    the    given    surflxces    be   the    pi-ojecting   cylinders   of    a 
curve  of  double  curvature,  then    will    Equations    (320)    become 

L  =  F{xs)=:  0; 
H^F'{yz)=0. 

And   because  L  is  now  independent  of  y,  and   H  is  independent  of  a;, 
we  have 

dH 


dy 
which  reduce  Equations  (324)  to 

df 


d  X 


=  0 


dL 

dx 


=  0; 


dt"  d  z  dz 


527) 


a&d  Equation    (325)  to 


\  d  t"  /       dz     ay 


+  (r-i/.5?!|)."4^ 

\  di^y      dx      d 

\  «r/       dx     dy  ^ 


^=  0. 


(328) 


This,    with   the   equations   of    the    curve,    will   give    the   j)lace   of   the 
centre   of  inertia   at   the   end   of  a   given   time. 

§  21G. — if  the  curve  be  plane,  the  co-ordinate  plane  a;  2,  may  be 
assumed  to  coincide  with  that  of  the  curve ;  in  which  case  the 
second  of  Equations  (327),  becomes  independent  of  y,  that  varia 
ble   reducing  to   zero,   and 


.     dll 

d-y  =  0,     and     —, — 
^  dy 


0; 


MECHANICS     OF    SOLIDS. 


223 


heuce  Equations    (327).  bcome 


,^    d'^  X  d  L 


}U^ 


..    d^z     ^    ^     dL    ^   ^,dH       ^ 
dt-  dz  dz 


r 


(329) 


and   because  the   factor 


^-^7l 


Equation    (328)    becomes,   on   dividing  out   the   common  factor 


dH 

di/ 


\  dt^/      dz         \ 


M 


d^ 
di 


0-^  =  «--(^««) 


§  217. — By    transposing  the  terms  involving  X,  in  Equations  (316) 
and    (318)  and  squaring   we   have 


A  (^7)^+  (^y+  m\ 


f     /  d^x\^  ^ 


H-(---?|)^ 


The  second  member  of  this  equation  is,-  Equation  (50),  the  square  of 
the  intensity  of  the  resultant  of  the  extraneous  forces  and  the  forces 
of  inertia.     Denoting   this   resultant   by  iV,  we  may  write  . 

V(^X+(^)  +  (^)^-^-  •••(-) 


and  dividing  each  of  the  equations 
dL 


X 


dL 


•224 


ELEMENTS     OF    ANALYTICAL    MECHANICS. 


obtained    by    the   transposition  just   referred   to,    by    Equation    (331), 
we  find, 


dL 
d  X 


X  -  M 


dj^ 

dL 


d^  X-) 
~dJ^ 


iV 


i/  •  -—- - 

d  i- 


N 


N 


d^  z 

dt" 


>•  (332) 


The  second  members  are  the  cosines  of  the  angles  which  the 
resultant  of  all  the  forces  including  those  of  inertia,  makes  with  the 
axes ;  the  first  members  are  the  cosines  of  the  angles  which  the 
normal  to  the  surface  at  the  body's  place  makes  with  the  same  axes. 
These  being  equal,  with  contrary  signs,  it  follows  not  only  that  the 
forces  whose  intensities  are 


vc^x+(^y+(^fr--> 


are  equal,  but  that  they  are  both  normal  to  the  surface,  and  act  m 
opposite  directions.  The  second  is  the  direct  action  upon  the  surface; 
the  first  is  the  reaction  of  the   surfiice. 

Equation  (331),  will,  therefore,  give  the  value  of  a  passive 
resistance  sufliciciit  to  neutralize  all  action  in  the  system  which  is 
inconsistent  will-,  the  arbitrary  condilion  in)j)0sed  uiu-.u  the  Ijody's 
path.  If  the  body  be  constrained  to  move  on  a  rigid  surface  oi 
line,    this   resistance    will    arise    from    its   roacliuu. 

§  218.— If  Equations    (332)    be    multiplied    by 
and   the   angles  whicii   the  iioimal    resistance  of  the  surface  makes  with 


MECHATflCS    OF    SOLIDS. 


225 


the   axes   x,  y,  z,    respectively,    be   denoted    by    ^^,    &^^   and    &^,   those 
equations  will  take   the   form 


Z  -  M-~  +  iV-cos^,     =0. 


(333) 


§  219. — To  impose  the  condition,  therefore,  that  a  body  in  motion 
shall  remain  on  a  rigid  surface,  is  equivalent  to  introducing  into 
the  system  an  additional  force,  which  shall  be  equal  and  directly 
oj)posed  to  the  pressure  upon  the  surface.  The  motion  may  then 
be  regarded  as  perfectly  free,  and  treated  accordingly.  The  same 
might  be  shown  from  Equations  (324)  to  be  equally  true  of  a 
rigid  curve,  but  the  principle  is  too  obvious  to  require  further 
elucidation. 

Equations  (333),  may,  therefore,  be  regarded  as  equally  appli- 
cable to  a  rigid  curve  of  any  curvature,  as  to  a  surface;  the  nor- 
mal reaction  of  the  curve  being  denoted  by  N,  and  the  angles 
which   N  makes   with   the   axes   x,  y,  z,   by    ^,,  &y,  and   &^. 

§  220. — To  find  the  value  of  N,  eliminate  d  t  from  Equations 
(333),  by  the   relation 

J_  _  JT 

dt    ~  ds' 

m   which   V  and  s  are  the  velocity  and   the  space ;   then  by  transpo- 
sition   these   equations   may    be    written 


iV^ .  cos  ^,  =  i/  •  F2  .  -^  -  X; 


iV'-cos^j,   =  M-  V 

iV^-cos^,    =21'  F2 
15 


2  .  tL 


ds^ 
~d? 


-  y; 

-  z. 


226  ELEMENTS     OF    ANALYTICAL    MECHANICS. 

Squaring,   adding   and   reducing   by   the  relations 

COS^^^  +   COS^^y     +   C0S2     ^^    z=    1, 


and   we   find 


IP  =   -l 


if  2  .  -^[^((/2^)2   +    (^2^)2   ^    (d^syl  4-   E2 


Resolving  E  into  two  components,  one  parallel  and  the  other  per- 
pendicular to  the  path,  the  former  will  he  in  equilibrio  with  the 
inertia  it  develops  in  the  direction  of  the  curve ;  and  denoting 
by    (p    the  inclhiation    of  R   to    the    radius    of  curvature,    we    have 


^  d  f  d  s2 


or, 


0  =  i?.sin(p  -M  .  F^ 


17' 


Squaring  and    subtracting    from    the  equation  above,  there  will  result. 


but 


if2 


X  d^x       Y  d^-y       Z    iPz        .        d^s\ 


-2M.F2.i2(^.-+-.^  +  ^^^, 


X   (?:?;       r  cr?/      Z    Jz 
R    d  s        K   da       K   d  s^ 


multiplying    the    second    member    by  p  -.-  p,  substituting   above,    and 
reducina;   by    the   relations, 

dx  dy                                     dz 

d"X     dx  drs        ds           c/2?/     dij  d^s  ds            d^z     dz  dh'        ds 

d  6-2      ds  da'^  ~    dis '           ds'^      dd  da'^  ds '           ds-     d.s  ds"^        ds ' 


/r  dj  dz    * 

X        ds        Y         ds       Z       '  ds 
cos9=-.p— +  --.p -—_  +  -.  p—-; 


R 


ds    '    R    '     ds   '   R 

"'St'c  Appendix  No.  2. 


ds 


MECHANICS     OF    SOLIDS. 


227 


and 


ds^ 


■^{pxf  +  {d'^yf  +  {pzf  -  {d?sf 


in    which    p    denotes    the    radius   of  curvature,    we   have, 


JV2 


iJ/2  •  -V  —  2 i2  cos  (p  -f-  m  cos2  (p ; 


and  taking  square  root, 


7.r          ^^  r> 

jy    r= K  COS  (p. 


(334) 


The  first   term    of  the   second   member  is, 

§  167,    the    centrifugal    force    arising   from 

the  deflecting  action  of  the  curve,  and  the 

last  term  is  the  normal  component  of  the 

resultant   R.     As  the   equation    stands,   its 

signs  apply  to  the  case  in  which  the  body 

is  on   the   concave    side  of  the    curve,  and 

the   resultant   acts    from  the   curve.     The  angle  9,  must   be   measured 

from  the   radius    of  curvature,  or   that   radius   produced,  according  as 

the  body   is    on    the  concave   or   convex    side  of   the   curve.      When 

the    body   is    moving    on    the    convex    side   of    the    curve,    the   first 

term    of    the    second   member    must    change    its    sign    and    become 

negative. 

§221.— Writing  Equations  (333)  under  the  form 


M 


df- 


Y  -\-  Nco^&, 


M~=  Z  -\-iYcosd,; 

multiplying  the  first   hy  2d z,  the  second  by  2di/,  the  third  by  2dz, 
adding  and  reducing  by  the  relation, 

,     /dx  ,         dy  ,  dz  \  ^ 

rf  s  (  --  •  cos  ^,  +  -^  .  cos  i3„   +  -—  •  cos  6,   )  ■  =  0, 
^d  s  ds  "         ds  y 


228  ELEMENTS     OF     ANALYTICAL     MECHANICS. 

the  second   factor    being   the   cosine    of  the   angle    made  by   the    noi- 
mal    and   tangent   to    the   curve,  we  have 

M  '  y ^^ ■ )  =':i[Xdx^ldij-{-Zdz)\ 

integrating  and  reducing  by 

dx~  +  dy"-  +  dz^ 

V  ^    ■::z:    i 

df^ 

vre  find 

M  F2  r=  2  f{Xdx  +   Ydy  +  Zdz)  +  C.      •     •     (335) 

This  being  independent  of  the  reaction  of  the  curve,  it  can  liave  no 
eiiect   upon    the    velocity. 

If  the    incessant   forces   be   zero,  then  will 

A"  =  0  ;     r  =:  0  ;    and    Z  =:  0  ; 

and 

,.  _  ^  . 
~  J/  ' 

that  is,  a  body  moving  upon  a  rigid  surfice  or  curve,  and  not  acted 
upon  by  incessant  forces,  will  preserve  its  velocity  constant,  and  the 
motion  will   be    uniform. 

We  also  recognize,  in  Equation  (335),  the  general  theorem  of 
the  living  force  and  quantity  of  work  ;  and  from  whicli,  as  before, 
it  appears  that  the  velocity  is  wholly  independent  of  the  path  de- 
scribed. 

Example  1. — Let  the  body  be  required  to  move  upon  the  interior 
surface  of  a  spherical  bowl,  under  the  action  of  its  own  Aveight.  In 
this  case, 

Z  =  .r-  +  y2  _f_  £.2  _  a2  ^  0  ;      .     .     .     •     (33G) 

dL        ^         dL        ^  dL        ^ 


o 


dx    ~  ~"''      dy    ~  "^'      d'z 


JNE  E  C  H  A X  I  C  S     OF     SOLIDS 


229 


ami   the   axis  of  z  being   vertical    and 
puiiitive    downwards, 

A"  1=  0  ;      F  =  0  ;     Z  ^  M  rj  ; 

which    values     in     Equations     (319), 
<rive 


y  •  ~^-^  —  X-  -A,  =  0 


1 


[ly  -y 


cP  y 


V.(337) 


dfi^'-tfi'-'' 


and  differentiating  the  equation  of  the 
sphere  twice,  we    have 

xd^-x  +  yd^y  +  z .  d"^ z  =  —  {dx^  +  dy"^  +  d z^~)  ; 

dividing   by  d P,  and   replacing   the    second  member  by  its  value    F^, 
the   velocity,  we  find, 

d'^z 


dfi 


But.  Equation  (335), 


d^  X  d"^  ?/ 


dt^ 


dfi 


=    -    F2. 


F2 


2y2  +  C 


(338) 


and  denoting  b}^  V  and  /•,  the  initial  values  of  V  and  0,  respectively, 
we   have 

F2=:    F'2   +   2^(.   -^0, 

wh'ch   substituted   above,  gives 

,.S  +,.^+,.^^2^(/.-.)-F'2    ..(339) 


(/^2 


^^2 


tZ/2 


Eliminate  ar,  y,  cZ^  _^^  ^-i  y^  from   this    equation    by    means  of  Equa- 
tions (336)  and  (337). 

From    the   latter  we  find, 

^C^2; 


d'^y        y  fd'^z         \ 

X    /d^-z  \ 


d'^x 


230     ELEMENTS  OF  ANALYTICAL  MECHANICS. 

which  substituted  in  Equation  (339),  and  reducing  by  means  of 
Equation  (336),  we  get 

multiplying  by  2dz,  and  integrating,  we  find 

a2.  ^  =  2y  {a^z  -  s3  +  k,2)  _  7.2  „2  4.  C; 

in  which  C  is  the  constant  of  integration,  and  to  determine  which, 
we  denote  the  component  of  the  velocity  V,  in  the  direction  of  the 
axis  2,  by  V^',   and   make   z  =  k.     This  being  done,  we  get 

(7  =  a2.F/2  _|.  7'2p  _  2^a2y5;; 
whence, 

a2.  ^  ^  2^  («22  _  ^3  ^  ^-22)  _    7/2^2   +   ^2  7^/2  _}_    p2  p  _   ^fjd'k, 

adding  and  subtracting  a?  V'^  in  the  second  member,  this   reduces  to 
a2.  ^  =  (a2  _  .2)  [F'2  _2<7  (^-  -  .)]  -  C, , 

in  which 

C,      =     (a2  -  yl-2)  7'2  _  «2  7-^/2_ 

Finding    the  value  of  d  t,  and  integrating,  we  have 

/adz  ,       , 

,  ■  '  '  '  (340) 

yfi^F^z^)  [ P2  _  2^  [k  -  z)\  -G, 

Could  this  equation  be  integrated  in  finite  terms,  then  won  id  z 
become  known  for  a  given  value  of  / ;  and  this  value  of  z  in 
Equation  (336),  and  the  first  of  Equations  (337),  after  integration, 
would  make  known  the  values  of  x  and  y,  and  hence  the  position 
of  the  body ;  its  velocity  would  be  known  from  Equation  (335). 
But    this    integration    is   not   possible. 


MECHANICS     OF    SOLIDS.  231 

§222. — We  may,  however,  approximate  to  the  result  when  the 
initial  impulse  is  small  and  in  a  horizontal  direction,  and  the  point 
of  departure  is  near  the  bottom  of  the  bowl.  Let  S  be  the  angle 
which  the  radius  drawn  to  the  variable  position  of  the  body  makes 
with  the  axis  of  s ;  (p,  the  angle  which  the  plane  of  the  angle  S 
makes  with  the  plane  through  the  axis  z  and  initial  place  of  the 
body,  supposed  in  the  plane  xz;  V  =  (3  ■\/ffa,  the  velocity  of  pro- 
jection in  a  horizontal  direction,  ^  being  a  very  small  quantity ; 
and   a   the   initial   value  of  6.     Then,  because  a  is  very  small, 

k  =  a  cos  a  =  a  (cos^  ^  a  —  sin^  ^a)  =  a  —  -^  a  a^  ; 
and  for  the  same  reason, 

z  =^  a  —  |-  a  .  fl2  •     also,     y  =  a;  tan  cp  ; 
F;2  =  0;      C,     =     [a2  _  a2  (1   -  1  a^-Y].f3^(fa  =  a^ ff ol^ (3\ 
after  neglecting  |-a*  in    comparison  with    unity. 


dt 

dt     dz 

dd  ~ 

~  dz     d&  " 

and    substituting   the    value    of  the   last  factor  from  Equation  (340), 

d  t  [Hi  6 

:U=-\ , •     •     (341) 

which  may  be  put  under   the   form 

li  t 


y  9  J  Vh? 


9     J    -y/(a2    —   /32)2    _    [2  ^2    —    (a?   +    /32)]2  ' 

whence   by   integration 


..=./..eos-.[?il^l^!+^)]+..     ..(3«) 


making    i  =  0,    and    ^  =  a,    we    have     C  —  —  cos   1  .  -y/a  -~  -y/g.,    or 
C  =  0 ;    and    solving    the   equation    uith    reference   to    ^,    we   get 

^2  _  1.  (^2  -f  ,/32)  +  1  (a2  _  /32) .  cos2  \/-  •  t      .  .  (343) 


232  ELEMENTS     OF     ANALYTICAL    MECHANICS. 

From  which  it  appears  that  the  greatest  and  least  values  of  & 
will  occur  periodically,  and  at  equal  intervals  of  time.  The  forme! 
of  these   values   is   found    by   making 

^=1;     Avhence  2\/'--t  =  0,     or  =  2'rr,     or  =:z  4:ir, 

and  so  on;  and  for  a  single  interval  between  two  consecutive  maxi- 
ma, without  respect   to    sign, 

'  =  '^X^-' (344) 

the  maximum  being  a. 

The   least   value   occurs   when 

cos  2  1  /-  •  if  =  —  1,     or  2  \  /-  •  ^  =  cr,  or  =^  3  "tt,    &c. 
V  a  V  a 

whence  for  a  single  interval  between  any  maximum  and  the  succeed- 
ing minimum, 

i  =  h'^\/^-^ (345) 

the    minimum    being    l3. 

The  movement  by  which  these  recurring  values  are  brought  about, 
is  called  oscillatory  motio)i ;  that  betw^een  any  two  equal  values  is 
called  an  oscillation ;  and  when  the  oscillations  are  performed  in 
equal    times,    they   are    said    to    be   Isochronous. 

Again, 

dcp         d  (p     d  t  ^ 
dJ  ~   dt'T^' 

substituting  for  — — ,  its  value  obtained  from  the  relation  y  =  r  tan  9, 

we  find 

dcp  1  /       dy  d X  \     dt 

dl   ~   a;2   +  y2    ■   V^  '  "7^  y  '  "77/    ■  Jl ' 

Integrating  the  first  of  Equations  (337),  wc  get 

d  y  d  X       ^  TT/  o        7 

at  at 


MECHANICS     OF    SOLIDS.  233 

substituting   this    above,  and  also  the    value    of  -— ,  given  by  Equa- 
tion (341),  we  find 

d  (p  a-  (3 

^  •  .     .    .    (346) 


dividing  this  by  Equation  (341), 
d  (p  /ff    a .  /3  /  ff  a .  f3 


dt         V    a       (J"  V    a 


\  {a?  +  /32)  +  4  [o?  -  /32) .  cos 2  a/^  •  t 


but 


q  q  .    ^      /  <l 

9*  I  :—-t  —  COS"  •  \  /  ^-  •  ^  —  sui-  \/'—-t\ 

a  V    a  V   a 


whence 

d  (p  fg  a  •  (3 


9      ,    ^    on      -o     /9 


^  +  p-^ .  sar\/  -^  •  t 
V    a  " 

from  which  we  find 

(5         ^'— •'^'^ 
a 


rf(p 


^'     t 


1  +---tan2^/-^  .^ 
a-  V    a 


(347) 


integrating,    and  taking    tangents    of  both    members, 

tan  ©  =  —  •  tan  -i  /  —  •  / (348) 

a  \     a 

from  which   the   azimuth    of  the   plane    of    osciUation    may   be    found 
at   the   end   of  any  time. 

Making   tan  (p  =  go,  we    have 

7,1  3  5        . 


234  ELEMENTS     OF     ANALYTICAL    MECHANICS, 

and   the   interval  from  the  epoch  to   the   first  azimuth  of  90°,  is 


1  /a 

^  =  —  *  •  \  /  — ' 


and   to   the   first   azimuth   of  270°, 

3  Fa 

and  the  interval  from  the  azimuth  of  90°  to  the  next  azimuth  of  270°, 


t..  —  t,  =  t  =  If  -x/ — J 
9 


equal   to   the   time  of  one   entire   oscillation. 

From    Equation    (348)    we    have,    after    substituting    for   tan  (p    its 
\ra.\\xe   in   the   relation   y  =  x  tan  9, 


^      =tan2A/^./; 


(3^  x^  V    a 

adding   unity  to   both   members, 

^;±-^=l  +  tanH/S.^; 
p2  a;2  V    a 


also   from   ?/  =  a; .  tan  (p, 

x^  +  y^ 


=  1  -f  tan2 


dividing   the  last  equation  by   this   one,  and  replacing  x'^  +  y^  by  its 
value  a^  —  z^  from    the    equation    of  the    sui-flice,  we  get 

1  +  tan2  y  —  •  t 
a^y^  +  /32..2  ^  132  .  («2  _  ,2)  .  ^_^^^^^^^^      '' ; 

but,  neglecting  the  term  involving  d*, 

a2  —  ^2  _  ^2  ^2  . 

substituting   this    above,  replacing    tan2q3    by    its    value    in    Equation 
(348),  and  6-  by  its  value   in  Equation  (343),  after  making 


\2\/ ^-  •  t  =  cos2  \  /  - 
V    a  V 


cos 2\/  —•  t  =  cos2  \ /  —  •  t  —  sin2  \ / —  •  t. 


MECHANICS    OF     SOLIDS. 


235 


and   reducing  by   the   relation, 


■^—  •  t  -{-  sin^ 
a 


t  =  1 


we   have 


_    -i-    ?L.    -      2 

ofi    ■+"  /32  -  "  5 


(349) 


which  shows  that  the  projection  of  the  path  of  the  body  on  the 
plane  xy,  is  an  ellipse  whose  centre  is  on  the  vertical  radius  of  the 
sphere,  and  that  '  the  line  connecting  the  body  with  the  centre  of 
the   sphere,  describes   a   conical    surface. 

If  a  =  /3,  then  will,  Equations  (343)  and  (348), 


t; 


^2    ^    a2    z=    /32;        (p    _    ./JL 

V   a 

and,  Equation    (349), 

«2    +    y2    _    £^2^2. ^g-Q^ 

hence,    the    body    will  describe    a    horizontal    circle    with    a    uniform 
niotion. 

The   pressure   upon  the    surface,  at  any  point  of  the    body's  path, 

is    given   by  the  value  of  N  in  Equation  (334). 

^2-2o.—-Exam23!e  2. — Let  the  body,  still  reduced  to  its  centre  of 
inertia  and  acted  upon  by  its 
own  weight,  be  also  repelled 
from  the  bottom  point  A 
of  the  bowl,  by  a  force  which 
varies  inversely  as  the  square 
of  the  distance  ;  required  the 
position  of  the  body  in  which 
it  would    remain  at   rest. 

As  the  body  is  to  be  at 
rest,  there  will  be  no  inertia 
exerted,  and  we  have 


'd¥ 


=  0 


(Py 
df 


-0; 


d^z 


=  0 


2ofi  ELEMENTS     OF     ANALYTICAL    MECHANICS. 

and  assuming  the  axis  z  vertical,  positive  upwards,  and  the  origin 
at   the  lowest  point  ^-1, 

X  =  a;2  +  y2  _^  .2  _  2^2  =  0,     •     <     •     .     (351) 

and  denoting  the  distance  of  the  body  fi-om  the  lowest  point  l)y  r, 
the  intensity  of  the  repelling  force  at  the  unit's  distance  by  L\  and 
the    force    at    any    distance   by   P,  then   will 

P  =  -^;     r  =    V^+y2  +  ,.2.   .     .     .     .     (352) 


X  XI  % 

fur    the   force  P,    cos  a  zz:    —  ;     cos  ,^  =  —  ;     cos  7  =  —  ;     for     tho 
weight  i/y,  cos  a'  =  0;     cos  /3'  =;  0  ;     cos  7'  =  —  1  ;    and 
Fx  Fv  Fz 

These    several  values  being  substituted  in  Equations  (319),  give 


(5_^,).,-^. (.-„)  = 


The  first  equation  establishes  no  relation  between  x  and  ?/,  since 
the  equilibrium,  which  depends  upon  the  distance  of  the  particle 
fiMiu  the  source  of  repulsion,  would  obviously  exist  at  any  point 
of  a  horizontal  circle  whose  circumference  is  at  the  proper  height 
from    the  bottom. 

From    the    second    equation    we  deduce, 


(353) 


Fa^W 

U 

F_  _  r3 
M  g  ~   a 


MECHA^MCS     OF  SOLIDS.  23t 

from   which  r  becomes    known ;    and    to  determine  the  position  of  the 

circle    upon    which    the    body    must    be  placed,  we    have,    by  makino- 
a;  =  0  in  Equations  (352)  and  (351), 


■y/z-  +  y"^  =  r, 
f"  +  z^-  —  2az  =  0. 

Equation  (353)  makes  known  the  relation  between  the  weight 
of  the  body  and  the  repulsive  force  at  the  unit's  distance;  the  in- 
tensity of  the  force  at  any  other  distance  may  therefore  be  deter- 
mined. 

If  there  be  substituted  a  repulsive  force  of  different  intensity, 
but  whose  law  of  variation  is  the  same,  we  should  have,  in  like 
manner, 

F'     _  r'3 
Mff   ~  V' 

hence, 

F :  F'  :  :  r^  :  r'^ ; 

that    is,  the    forces    are  as   the  cubes    of  the    distances    at   which    the 
body   is    brought   to   rest. 

If,  instead  of  being  supported  on  the  surface  of  a  sphere,  the 
body  had  been  connected  by  a  perfectly  light  and  inflexible  line 
with  the  centre  of  the  sphere  and  the  surface  removed,  the  result 
would  have  been  the  same.  In  this  form  of  the  proposition,  we 
nave    the  cominon  Electroscope. 

The  differential  co-efficients  of  the  second  order,  or  the  terms  which 
measure  the  force  of  inertia,  being  equal  to  zero.  Equations  (->o"2), 
show  that  the  resultant  of  the  extraneous  forces,  in  this  case  the 
weight  and  repulsion,  is  normal  to  the  surface,  which  should  be  the 
case ;  for  then  there  is  no  reason  why  the  body  should  move  in 
one  direction  rather  than  another.  The  pressure  upon  the  surface  is  q 
given    by    the    value    of  jV,  in  Equation    (334).  "^     | 

g  2'24. — Exiiiiipk  3.     Let   it   be    required  to  find  the  circumstance.s 


238 


ELEMENTS     OF    ANALYTICAL    MECHANICS. 


of  motion   of  a  body   acted   upon   by   its   own   weight  while   on   the 
arc  of  a  cycloid,  of  which 
the   plane   is    vertical,  and 
directrix  horizontal. 

Taking  the  axis  of  0, 
vertical;  the  plane  zx,  in 
the  plane  of  the  curve; 
and  the  origin  at  the  low- 
est  point,  then   will 


=  X  —\/2a 


2  •      -^     ^  A 

z  —  z^  —  a  versin     —  =  0 ; 
a 


in   which  z   is   taken  positive   upwards. 


dL 
d  X 


=  1 


dL 

u 


=-/- 


X  =  0  ;     Z  =  -  Mg, 

and  Equation  (330)  becomes 


d"  X 

df 


2  a  —  z  d"^  z 

^-+'  +  -d¥  =  '' 


(354) 


(355) 


(356) 


and   by   transposition    and   division, 


d^x 


dt~ 


l-l  a  —  2       d  t~       r2  «  - 
From    the    equation    of  the    curve    we    find. 


(3.57) 


/2  a  —  z 
2dz  =  2rfs-W ; — ; 


(35S) 


multiplying    by    Equation    (357),  there    will    result 


2dx.d^x  ,         2dz.d^z 

n  -2f/dz- 


dfi 


dt^ 


MECHANICS     OF    SOLIDS.  239 

and    by    integration, 


or, 

dfi 


F2=  0-2gz) 


and    supposing    tlie    velocity  zero,   when   z=.h-j 

0=  C-)lgh; 
which   subtracted   from    the   above  gives 
dx^  -\-  dz 


^2ff{h-z)', (^59) 


di^ 
and    eliminating    dx^   by    means    of  Equation   (358), 


—-  'ihz  —  z^\ 

dfi        a    ^  ' 


whence, 

dz 


dt 


s/l- 


9    ^/hz  —  z^ 


the    negative    sign    being    taken   because   «    is   a  decreasing    function 
of   t. 

By   integration, 

/a       P         dz  '  /a  .  -1  2z        „ 

t  =^  —  \  /  —  •  /  — .  =  —  \  / versui     •  — h  C, 

\   9    -f    ^hz  -z^  y   9  A 

Making  z  =  h,  wq  have 

0  1=  —  \  /—  •  versin"  ^  2  +  C  ; 


UO  ELEMENTS  OE  ANALYTICAL  MECHANICS. 


AMionce, 


C  =    T!  \l J 

fl 


and 


^-Vy  (*-^ersm    \?l) (360) 

When    the   body  has  reached  the  bottom,  then  will  2  :^  0,  and 


■which  is  wholly  independent  of  /i,  or  the  point  of  departure,  and 
we  hence  infer  that  the  time  of  descent  to  the  lowest  point  will  be 
the  same  in  the  same  cycloid,  no  matter  from  what  point  the  body 
'Starts. 

Whenever    z  =  A,    the   body    will,    Equation    (359),  stop,    and    we 
shall    have    the  times   arranged  in  order   before    and    after  the    epoch, 


the    diflerence   between  any  two    consecutive  values   being 

2*\/  — 
V    ij 

The   body    will,    therefore,    oscillate   back    and    forth,  in    equal    times. 

The    cycloid  is,  on    this    account,  called  a  Tautochronous  curve. 
The    pressure    u])on    the    curve   is   given    by   Equation  (334), 
The  time  being  given  and  substituted  in  Eipiation  (3G0),  the  value 

of  z   becomes    known,  and   this,  in    E(|uatiuns    (359)   and    (358),  will 

give   the   body's  velocity  and   plftce. 

§225. — Exum-ple  4. — Let  a  body  reduced  to  its  centre  of  inertia, 
and  whose  wpight  is  denoted  by  PF",  be  supported  by  the  action 
of  a  constant  force  upon  the  branch  E II  of  an  liyprrhola,  of  which 
the  transverse  axis  is  vertical,  the  force  being  directed  to  the  centre 
of  the   curve.     Required    the   position  t)f  equililjriuin. 


MECHANICS     OF     SOLIDS. 


241 


Denote  the  constant  tcrce  by  W,  which  may  be  a  weight  at  the 
end  of  a  cord  passing  over  a  small  wheel 
at  0,  and  attached  to  the  body  M.  De-  ^ 
note  the  distance  CM  by  r,  and  the  axes 
of  the  curve  by  A  and  B.  Take  the  axis 
s  vertical,  and  the  curve  in  the  plane  xz. 
Make 

F'  =  W, 
P"  ~  W 


then  will 


cosy'  =:  1,     cos  a'  =  0, 


cosy"  = 5     cos  a'  = 5 

r  r 


X  -  P'  cos  a'  +  P"  cos  a" 


W 


Z  =  P'  cos  y'  +  P"  cos  y"  =  W  —  W 


V       \i^U 


d 


and   as   the   question   relates  to  the   state   of  rest,/ )(-Lit~4^i^  j  Zt-i  "  ^ 


The   Equation   of  tho   curve   is 

whence, 

dL 


7 


dx 


2A^x, 


dz 
these  values  substituted   in  Equation  (330),  give 

W'B^  —  -  WA^x  +  W'A^  —  =  0; 


whence, 


{A^  H-  B^)  W'-z  -  WA^r  =  0 
16 


(301) 


242  ELEMENTS     OF     ANALYTICAL     MECHANICS. 

But 

r2  =  a;2  +  22  ^  g2  +  —  z'  -  B^^z' ^^ ^2 j 

whence,  denoting  the  eccentricity  by  e, 


r   ==  V^2g2   _    ^2 

and   this,    in   Equation    (361),  gives  after  reduction, 
-  ^  ■  TF" 

which,  with  the  equation  of  the  curve,  will  give  the  position  of 
equilibrium. 

If  W  e  be  greater  than  TF,  the  equilibrium  will  be  impossible- 
If  We  =   IF,  the   body  will  be  supported  upon  the  asymptote. 

The  pressure  upon  the  curve  is  given  by  Equation  (834). 

§  22(5. — Exarn'ple  5. — Required  the  circumstances  of  motion  of  ^t 
body  moving  from  rest  under  the  action  of  its  own  weight  up(.)n  an 
inclined    right   line. 

Take  the  axis  of  z  vertical, 
the  plane  z  x  to  contain  the 
line,  and  the  origin  at  the 
point  of  departure,  and  let  z 
be  reckoned  positive  down- 
wards.    Then  will 


L  =  z 


a  X 


0, 


d  L 

dz 


=  1 


dL 

d  X 


z 

A 

1 

^" 

\ 

x\ 

D 

H 

'\^ 

y 

"y 

rj 

C 

which    in  Equation    (330)  give,  after  omitting  the    ionimon  factor  J/, 


rf2.r 

dl~  ^ 


d"—. 


dt 


V  -=  0. 


(S02) 


From  the  equation  of  the  line  we  have 

d"  z 


MECHANICS     OF    SOLIDS.  243 


which  in  Equation  (302),  after  slight  reduction, 
(Pz  _       gg 

Multiplying  by  2dz,  and   integrating, 

fhe   constant    of  integration    being   zero. 
Whence 


and 


V       g-d'         2t/z 

,  =  ^s^zr;    /i(L+^)...,  .  .  .(308) 

V         a  6t"  \        q  a?z  ^       ' 


the  constant   of  integration  being  again  zero. 

The   body  being   supposed   at  i?,  then  will   z  =  AD\    and    if  we 
draw  from  B  the  perpendicular  B  C  to  A  B^  we  have 


AB"^        1  +  a^  ,\\ 


,2  ' 


which  substituted  above, 


-      ^^^ 


-.   ttri 


in    which  d  denotes  the  distance  A  C. 

But  the  second  member  is  the  time  of  falling  freely  through  the 
vertical  distance  d ;  if,  therefore,  a  circle  be  described  upon  A  C  as 
a  diameter^  we  see  that  the  time  down  any  one  of  its  chords,  ter- 
minating at  the  upper  or  lower  point  of  this  diameter,  will  be  the 
same  as  that  through  the  vertical  diameter  itself  This  is  called  the 
mechanical   property  of  the  circle. 

Exowple  0. — A  spherical  body  placed  on  a  plane  inclined  to  the 
horizon,  would,  in  the  absence  of  friction,  slide  under  the  action  of 
its  own  weight ;  but,  owing  to  friction,  it  will  roll.  Required  the 
circumstances  of  the    motion. 


^^^+*'  i-"^ 


2U 


ELEMENTS     OF    ANALYTICAL     MECHANICS. 


If  the  sphere  move  from  rest  with  no  initial  impulse,  the  centre 
will  describe  a  straight  line 
parallel  to  the  element  of 
steepest  descent.  Take  the 
plane  xz,  to  contain  this 
element,  the  axis  z  vertical 
and  positive   upwards. 

The  equation  of  the  path 
will   be, 


L  =  z  -\-  X  tan  a  —  h  =  0  ; 


whence, 


dL 


=  1 


dL 
d  X 


=  tan  a. 


The  extraneous  forces  are  the  weight  of  the  sphere  and  the  fric- 
tifUi.  Denote  the  first  by  W,  and  the  second  by  F.  The  nature 
of  friction  and  its  mode  of  action  will  be  explained  in  the  proper 
place,  §  354 ;  it  will  be  sufficient  here  to  say  that  for  the  same 
weight  of  the  sphere  and  inclination  of  the  plane,  it  will  be  a  con- 
stant force  acting  up  the  plane  and  opposed  to  the  motion.  We 
shall    therefore    have 

Z  =  —  Mg  +  ^sin  a;     X  —  —  Fcosa, 

which   values,  and   those   above   substituted    in    Equation  (330),  give 


—  F  cos  a 


, ,    d"  X  -        / 


Fsuia  4-  AT-  '4-^1  ';^»  a  =  0. 

d  t^y 


But  from   the   equation  of  the   path,  we   have 
d"Z  =  —  d"^  X  •  tan  a  ; 
and    eliminating   d^  x  by  means  of  this   relation,  there   will   result 


d^z 
df 


=  sm  a  K-Tj.  —  g  sm  a) 


MECHANICS     OF    SOLIDS.  245 

Multiplying    by    2dz^    iiitcgnitiiig    and    maicing    the    velocity    zero 
when  z  —  h,  we   have 

-^  =  F2  ^  2  sin  a(~-ff  .sill  .)  •  {z  -  h). 

This   gives 

1  :l  Z 

dt  = 


aiid    by  intogi'ation,  the    tinae  being  zero  when  z  —-  A, 

A  —  s  =  I-  sin  a  (^  •  sin  a j~  )  •  t"^.       .        .        .      ('''). 

Again,  all  axes  in  the  sphere  through  its  centre,  arc  princi])al 
axes ;  the  sphere  will  only  rotate  about  the  movable  axis  y,  in 
which  case  v^  and  v^  will  each  be  zero,  and  Equations  (202)  will  give 

-47  =  -- 


wherein, 


'   '       dt  dt^  ^         ' 


r  being   the    radius    of  the    sphere. 
Whence, 


f/2^  Fr 


d  t-         Mk^ 

Multiplying    by   2(^4,  integrating,  and    maldng    the    angular  velocity 
and  the   arc  4^  vanish   together, 

dj^  _2Fr 

whence, 

fMk}~   d-^' 
d  t  —  ■   '  


./Ml. 


246  ELEMENTS     OF     ANALYTICAL    MECHANICS, 

and   by    integration,  making  t  and  ■\^  vanish  together, 

F.r 

^        ^  J/.  A.-,2 

Also,    because    the    length  of   path   described  in   the    direction   of  the 
plane  is  r.4^,  we  have,  in  addition, 

h  —  2  =  r  .  4'  •  sin  a ; 

and     eliminating  ^  from    this    and    the    above    equation,    there    Avill 
result 


V  jP  .  r^ .  sui  a  ^  ' 

Dividing    Equation  (a)   by   Equation    (6),   and  solving   with    respect 
to    F, 

and  this  in    Equation    [b),   gives 


V    i7  •  snr  a  r^  ■;  v 


If   the    sphere    be   homogeneous,   then    will 


k\  =  I  r\  and  t  =  . /^l^^^  .  ./T. 

'        ^  V    ^- sill"  a      V  5 

if   the    matter    be   all    concentrated    into    the    surface,  then    will 


i-  =  |,-  and  t^J-ii!^=J).Jt 

which    times   are   to   one   another   as    -v/ST  to    -a/So! 


CONSTKAINED   MOTION  ABOUT   A   FIXED   POINT. 

§  227.— If  a  body  be  retained  by  a  ^xed  point,  the  fixed  and 
what  has  been  thus  flir  regarded  as  a  movable  origin  may  both  be 
taken  at  this  point;  in  which  case,  S x^,  fly^,  Sz^,  in  Equation  (40), 
will  be  zero,  the  first  three   terens  of  that  general    equation    r^f  equi- 


MECHANICS     OF    SOLIDS.  247 

librium  will  reduce  to  zero  independently  of  the  forces,  and  the  equi- 
librium will  be  satisfied  by    simply  making 

2  P  (ar  cos  /3  —  y  cos  a)  —  2  m — =  0  : 


z  F  [z  cos  a  —  X  cos  y)  —  2  m  • =  0 


d(^ 

2. 

d" 

X    —   Z 

d'^z 

dfi 

?/ 

rf2 

z  —  z . 

d^y 

^  P{y  cosy  —  z  cos  (3)  —  i:  m  ■  — -— ■-  =  0 


(3G5) 


the   accents   being    omitted    because    the    elements    ?«,  m',  &e.,    being 
referred    to    the    same    origin,  x\  y\  z'  will  become  x,  y,  z. 

The  motion  of  the  body  about  the  fixed  point  might  be  discussed 
both  for  the  cases  of  incessant  and  of  impulsive  forces,  but  the  discus- 
sion being  in  all  respects  similar  to  that  relating  to  the  motion  about 
the  centre  of  inertia,  §  127  and   §  IVS,  we  pass  to 

COJS'STKAINED   MOTIOX   ABOUT   A   FIXED    AXIS. 

§228. — If  the  body  be  constrained  to  turn  about  a  fixed  axis, 
both  origins  may  be  taken  upon,  and  the  co-ordinate  axis  y  to 
coincide  with  this  axis;  in  which  case  Sx^,  ^Vi^  Sz^,  Sep  and  StS, 
in  Equation  (40),  will  be  zero,  and  to  satisfy  the  conditions  of 
equilibrium,  it  will  only  be  necessary  for  the  forces  to  fulfil  the 
condition, 

z  d   V  X  •  d'^  "^ 

2  P{z  cos  a  —  X  cos  7)  -  2  m — -^ =  0    •  •  (3G6) 

the    accents  being    omitted  for  reasons  just  stated. 

§229. — The   only  possible    motion    being    that    of    rotation,    let    OS 
transform  the   above    equation    so    as  to  contain  angular  co-ordinates. 
For  this   purpose  we   have.  Equations  (36), 

«'  =  ?•"  sin  .4. ;     2'=:r"cos4. (307) 

ia  which  r"  denotes  the   distance    of  the  element  m  from  the  axis  y. 
Omitting  the  accents,  diflerentiating  and  dividing   by  d  t,  we  have 

dx  .  d-L       dz  ,     .     d-1^  ,..  ..v 

—  =  r  cos  4/  — -^ ;     —  =  —  ?•  sm-1 .  -~  •     •     •      308^ 
dt  ^  dt        dt  ^     dt  ^        ' 


24:8     ELEMENTS  OF  ANALYTICAL  MECHANICS. 

Now, 

1  /      dx  dz\ 

It         \       dt  dt/   ^ 


Z'd"^  X        X-  d"Z 


d  f'  d  fi  d 

whence  by  substitution,  Equations  (307)  and  (oG8), 

d^-x  d^"Z  ]       ,  /  ,    c/^\  ^    c?H 


-    dt        \        dt/ 


^"d¥      ""'  di-'  ~  dt      V  *  dt)  -''"    dt''  ' 

and   since    — —    must  be  the  same  for  every  element,  we  have,  Equa^ 
dt^ 

tion    (366), 

Im  r- '  —r-TT  =  2  P  (2  cos  a  —  x  cos  7), 
dfi  ^  '' 


and 


d'^  \         1^  P  '  {z  cos  a  —  x  cos  7) 


(309) 


That  is  to  say,  the  angular  acceleration  of  a  body  retained  by  a 
fixed  axis,  and  acted  upon  by  incessant  forces,  is  equal  to  the 
moment  of  the  impressed  forces  divided  by  the  moment  of  inertia 
with   reference   to   this  axis. 

Denoting  the  angular  velocity  by  Fj ,  and  the  moment  of  inertia 
by  /,  we  find,  by  multiplying  Equation  (369)  by  2f/4'  ^^'^^  integrating, 

/Fi2  r=  ^f^  P{z  cos  a  —  xcosy)d-:^  +  C, 

and   supposing   the   initial  angular   velocity  to  be   F/,  we  have 

/(  F,2  -   r/2)  =,  2  y  V  p  (.  cos  a  -  X  cos  7)  d-\.. 

But  the  second  member  is,  §  107,  twice  the  quantity  of  work 
about  the  fixed  axis ;  whence  the  quantity  of  work  performed  be- 
tween the  two  instants  at  which  the  body  has  any  two  angular 
velocities,  is  equal  to  half  the  difference  of  the  squares  of  these 
velocities  into  the  moment  of  inertia,  or  to  half  the  I'ving  force 
gained    or   lost    in    the   interval. 


MECHANICS     OF    SOLIDS. 


2-i9 


If  Fj^  —  F/^  =  1,  we  find  the  value  of  I  to  be  twice  the 
quantity  of  woi'k  required  to  produce  a  change  in  the  square  of  the 
angular  velocity  equal   to   unity. 


COMPOUND    PENDULUM. 

§  230. — Any    body    suspended  froni.    a   horizontal    axis  A  B,    about 
which    it    may    swing  with  freedom    under    the 
action  of  its  own  weight,  is  called  a  comjMtoid 
pendulum. 

The  elements  of  the  pendulum   being  acted 
upon    only  by  their  own  weights,  we  have 

P  =1  mg;     P'  =:  J'l-' ff,  &G-  ; 

the  axis  of  z  being  taken  vertical  and  positive 
downwards, 

cos  a  :=  cos  a'  =  &c.  =  0  ; 

cos  V  =  cos  y'  =:  &c.  =  \, 

and  Equation  (369)  becomes 


~d^ 


2  m  X 


=  -  9 


(370) 


Denote  by  e,  the  distance  A  G,  of  the  centre  of  gravity  from   the 
axis;    by  4^,    the    angle  H A  G^  which 
xi  G  makes  with  the  plane  yz;    by  .r^, 
the    distance  of  the    centre  of  gi'avity 
from  this  plane ;     then   will 

ar^  =  e  .  sin  4^  ;  / 

and  from  the   principles  of  the  centre 
of  gravity, 

2  m  X  —  31  x^  =  M.  e  .  sin  4^ ; 

which  substitute  1  above,  gives 


£^24, 


=  —  0 


M .  e  .  sin  4^ 
2  111  r"^ 


(371) 


250  ELEMENTS     OF     ANALYTICAL    MECHANICS. 

Multiplying  by  2d-]y,  and  integrating, 

Denoting  the  initial  value  of  4^  by  a,  we  have 


Me 
0  =  2g-- 5 -cos  a  +  C; 


whence, 
but 


dV-        ^        M.e     .        . 

■T^  =^9-^--A^o^-\^  -  co^a);    .     .     .     .(372; 


a-  a* 

COS  a  =  1  —  — —  +   -    ^,    .,     ,  —  &c. 
1.2  1 . 2    o . 4 

and  taking  the  value  of   y,  so    small    that  its    fourth   power    may  be 
neglected  in  comparison  with  radius,  we  have 

COS  4-  —  cos  a  — ; 

which  substituted  above,  gives,  after  a  slight  reduction,  and  replacing 
2  TO  r^  by  its  value  given  in  Equation  (216), 


a,-     .A'^  +  ^' 


^4. 


e.g        i^_^ 


the  negative  sign  being  taken    because  4*  ^s  a    decreasing  function   of 
the  time. 

Jntegratinc,  we  have 


<  =  \/-^— ^— -cos      — (373) 

\       e  .<j                    a  ^        ' 

The    constant    of    integration  is   zero,  because  when  -vj^  =  a,  we  have 

t    z=z    0. 


MECHANICS    OF    SOLIDS,  251 

Making  vj^  =  —  "-j  '^^'^  have 


'  =  W^^' (SM) 

which  gives  the  time  of  one  entire  usciHation,  and  from  which  we 
conclude  that  the  oscilhitions  of  the  same  pendulum  will  be  isochro- 
nal, no  matter  what  the  lengths  of  tiie  arcs  of  vibration,  provided 
they  be   small. 

If  the  number  of  oscillations  performed  in  a  given  interval  say 
(en  or  twenUj  minnies,  be  counted,  the  duration  of  a  sino-le  oscillation 
will  be  found  by  dividing  the  whole  interval  by  this  number. 

Thus,  let  6  denote  the  time  of  observation,  and  JV  the  number  of 
oscillations,  then  will 


==^=^"V- 


'  K'  + 


e.ff 


and  if  the  same  pendulum  be  made  to  oscillate  at  some  other  location 
during  the  same  interval  S,  the  force  of  gravity  being  different,  the 
number  iV'  of  oscillations  will  be  different ;  but  we  shall  have,  as 
before,  ff'  being  the  new  force  of  gravity, 


X-,2  4.  e2 


iV  V       e.  cj' 

Squaring   and  dividing  the  first  by  the  second,  we  find 

that  is  to  say,  the  intensities  of  the  force  of  gravity,  at  different 
places,  are  to  each  other  as  the  squares  of  the  number  of  oscilla- 
tions performed  in  the  same  time,  by  the  same  pendulum.  Hence, 
if  the  intensity  of  gravity  at  one  station  be  known,  it  will  be  easy 
to  find  it   at   others. 

§  231  .—From  Equation    i^S),  we  have 


dt^ 


2  m  r^  =  2  IT.  ff  .  e  (cos  -^  —  cos  a)  ;     •     •     (375) 


252  ELEMENTS     OF     ANALYTICAL    MECHANICS. 

and  making 

—^  z=    V,:     2mr2  =  /:     e(cos%L  —  cos  a)  =  //; 

we  have 

I.V,^  -"IM.g  .H-, (3TG) 

in  which  //,  denotes  the  vei'tical  height  passed  over  hy  the  centre 
of  gravity,  and  from  which  it  appears  tliat  the  penduhim  will  come 
to  rest  whenever  ^  becomes  et[ual  to  a,  on  either  side  of  the  ver- 
tic;d    plane  through    the    axis. 

^  202. — If  the  whole  mass  of  the  pendulum  be  conceived  to  be 
concentrated  into  a  single  poirit,  the  centre  of  gravity  must  go 
there  also,  and  if  this  point  be  connected  with  the  axis  by  a  medium 
without  weight  and  inertia,  it  becomes  a  majile  ^5e«tZi(^M??z.  Delic- 
ti ng  the  distance  of  tlie  point  of  concentration  from  the  axis  by  Z, 
v,(j   have 

A:^  =  0 ;     e  =  /, 

/ 

which  reduces  Equation  (374)  to  /  -  ^<3  j^ 

rj-  5z.^  /o£ 

'  =  -\/7 (''''> 

■-\rx 

If  the  point    be  so  chosen  that  V    5v 


I   _     /i-;^  +  c2 
9  ~y     <i-y 


or, 


/^^_±_^; (378) 


tlu'  simple  and  compound  pendulum  will  perform  their  oscillations  in 
the  same  time.  The  former  is  then  called  the  equivalent  simple  pen- 
di'litii/ ;  and  the  point  of  the  comjiound  pendulum  into  which  the 
mass  may  be  concentrated  to  satisfy  this  condition  of  equal  duration, 
is  called  the  centre  of  oscillation.  A  line  through  the  centre  of 
oscillation  and  parallel  to  the  axis  of  suspension,  is  caUed  an  axis  of 
<jsciliiUion, 


MECHANICS     OF    SOLIDS.  253 

§  233. — The  axes,  of  oscillation  and  of  suspension  are  reciprocal. 
Denote  the  length  of  the  equivalent  simple  pendulum  when  the  com 
pound  pendulum  is  inverted  and  suspended  from  its  axis  of  oscillation, 
by  I',  and  the  distance  of  this  latter  axis  from,  the  centre  of  gravity 
by  e',  then  will 

I   =  e  -{-  e'     or     e'  =  I  —  e ; 

and,  Equation  (378), 

e'         ~  l-e         ' 

and  replacing  /,  by  its  value  in  Equation  (378),  we  find 


That  is,  if  the  eld  axis  of  oscillation  be  taken  as  a  new  axis  of  sus 
pension,  the  old  axis  of  suspension  becomes  the  new  axis  of  oscilla- 
tion. This  furnishes  an  easy  method  for  finding  the  length  of  an 
equivalent  simple   jDendulum. 

Difierentiating  Equation  (378),  regarding  I  and  e  as  variable,  we 
have 

de  e^ 

and  if  Z   be   a   minimum, 

de~  e2 

■whence, 

e  ^l-^ . 

But  when  I  is  a  minimum,  then  will  t  be  a  minimum,  Equa- 
tion (377).  That  is  to  say,  the  time  of  oscillation  will  be  a 
minimum  when  the  axis  of  suspension  passes  through  the  principal 
centre  of  rjyration^  and  the  time  will  be  longer  in  proportion  as  the 
axis  recedes  from    that    centre. 


25i 


ELEMENTS     OF     ANALYTICAL    MECHANICS. 


\ 


l^et  A  and  A'  be  two  acute  parallel  prismatic  axes  firmly  con- 
nected with  the  pendulum,  the  acute  edges 
being  turned  towards  each  other.  The 
oscillation  may  be  made  to  take  place 
about  either  axis  by  simply  inverting  the 
pendulum.  Also,  let  M  be  a  sliding  mass 
capable  of  being  retained  in  any  position 
by  the  clamp-screw  If.  For  any  assumed 
position  of  M,  let  the    principal    radius  of  / 

gyration    be    G  C;    with    G    as    a    centre,  j 

G  C  as   radius,  describe    the  circumference  \ 

CSS'.  From  what  has  been  explained, 
the  time  of  oscillation  about  either  axis 
will    be    shortened    as    it    approaches,    and 

lengthened  as  it  recedes  from  this  circumference,  being  a  minimum, 
or  least  possible,  when  on  it.  By  moving  the  mass  M,  the  centre 
of  gravity,  and  therefore  the  gyratory  circle  of  vrhich  it  is  the 
centre,  may  be  thrown  towards  either  axis.  The  pendulum  bob  being 
made  heavy,  the  centre  of  gravity  may  be  brought  so  near  one  (>f 
the  axes,  say  A',  as  to  place  the  latter  within  the  gyratory  cir- 
cumference, keeping  the  centre  of  this  circumference  between  the 
axes,  as  indicated  in  the  figure.  In  this  position,  it  is  obvious  that 
any  motion  in  the  mass  M  would  at  the  .same  time  either  shorten 
or  lengthen  the  duration  of  the  oscillation  about  both  axes,  but 
unequally,  in  consequence  of  their  unequal  distances  from  the  gyratory 
circumference. 

The  pendulum  thus  arranged,  is  made  to  vibrate  about  each  axis 
in  succession  during  equal  intervals,  say  an  hour  or  a  day,  and  the 
number  of  oscillations  carefully  noted;  if  these  numbers  be  the 
same,  the  distance  between  the  axes  is  the  length  /,  of  the  equiva- 
lent simple  pendulum  ;  if  not,  then  the  weight  M  must  be  moved 
towards  that  axis  whose  number  is  the  least,  and  the  trial  repeated 
till  the  numbers  are  made  equal.  The  distance  between  the  axes 
may  be    measured  by  a  scale  of  equal  parts. 

§234. — From  this  value  of  I,  we  may  easily  find  that  of  the  simple 
tecon<rs   2-^e/irf»/«?/i ;    that  is  to   say,  the  sin^ple  pendulum  which  will 


MECHANICS     OF     SOLIDS.  255 

perform  its  vibration  in  one  second.  Let  N,  be  the  number  of 
vibrations  performed  in  one  iiour  by  the  compound  pendulum  whose 
equivalent  simple  pendulum  is  I ;  the  number  performed  in  the 
same  time  by  the  second's  pendulum,  whose  length  we  will  denote 
by  l\  is  of  course  3600,  being  the  number  of  seconds  in  1  hour, 
and  hence, 

—  =  T  =  If  v/— , 

N  \   g 

==  r  =  *     ' 


3600^  V   g 

and  because  the  force  of  gravity  at  the  same  station  is  constant, 
we  find,  after  squaring  and  dividing  the  second  equation  I)}'  the  first, 

'=4^^ (^''" 

Such  is,  in  outline,  the  beautiful  pi'ocess  by  which  Kater  determined 
the  length  of  the  simple  second's  pendulum  at  the  Tower  of  London 
to    be   39,13908  inches,  or  3,26159  feet. 

As  the  force  of  gravity  at  the  same  place  is  not  supposed  to 
change  its  intensity,  this  length  of  the  simple  second's  pendulum 
nmst  remain  forever  invariable ;  and,  on  this  account,  the  English 
have  adopted  it  as  the  basis  of  their  system  of  weights  and  measures. 
For  this  purpose,  it  was  simply  necessary  to  say  that  the  •g-.-g-^Ts-g"' 
part  of  the  slurple  seconcVs  j^endulmn  at,  the  Toioer  of  London  shall 
be  one  English  foot,  and  all  linear  dimensions  at  once  result  from 
the  relation  they  bear  to  the  foot ;  that  the  gallon  shall  contain 
i'tW"  ^^  ^  C"^'C  ^"^t,  and  all  measures  of  volume  are  fixed  by  the 
relations  which  other  volumes  bear  to  the  gallon  ;  and  finally,  that 
a  cubic  foot  of  distilled  water  at  the  temperature  of  sixty  degrees 
Fahr,  shall  weigh  one  thousand  ounces,  and  all  weights  are  fixed  by 
the    relation  they   bear   to    the  ounce.  ■ 

§235.— It  is  now  easy  to  find  the  apparent  force  of  gravity  at 
London  ;  that  is  to  say,  the  force  of  gravity  as  affected  by  the  cen- 
trifup-al  force  and  the  oblatcness  of  the  earth.     Tlie  time  of  oscillation 


256  ELEMENTS     OF     ANALYTICAL    MECHANICS. 

Ijcing    one   second,  and    the    length  of  the    simple   pendulum    3,2G159 
feet,  E(|uation  (377)  gives 


/3,2G159 
9        ' 
whence, 

^  =  ^2  (3  0G159)  ^  (3,141G)2.  (3,2G159)  =  32,1908  feet. 

From  Equation  (377),  we    also  find,  by  making    t  one    second, 


and   assuming 


we  have 


I  ^  X  ■{-  y  cos  2  -.]^, 


i-^^  X  -\-  y  cos  24. (380) 

Now  starting  with  the  value  for  g  ■  at  London,  and  causing  the 
same  pendulum  to  vibrate  at  places  whose  latitudes  are  known,  we 
obtain,  from  the  relation  given  in  Equation  (374)',  the  corresponding 
values  of  g,  or  the  force  of  gravity  at  these  places ;  and  these 
values  and  the  corresponding  latitudes  being  substituted  successively 
in.  Equation  (380),  give  a  series  of  Equations  involving  but  two  un- 
known quantities,  which  may  easily  be  found  by  the  method  of 
least   squares. 

Tn  this    way   it   has   been   ascertained  that 

"ffS.a;  =  32,1808     and     ^^^^  _  _  o,OS21  ; 

whence,  generally, 

^  =  32,1808  -  0,0821  cos  2  v}.;      .     .     .-   .     (381) 

and   substituting   this   value    in    Equation    (377),  and    making    ^  ==  1, 
we  find 

I  =  3,20058  -  0,008318  cos  2  4.     •     •     •     •     (382) 

Such   is   the   length  of   the   simple   second's    pendulum   at   any   place 
of  which   the   latitude  is  ■^. 


MECHANICS     OF     SOLIDS. 


257 


If  wo  make  -^  =  40°  42'  40",  the  latitude  of  the  City  Hall  of 
New  York,  we    shall  llnJ 

I  z^  3,->r)938  =  39,"ll256. 

§230. — The  principles  which  have  just  been  exjdained,  enable  us 
to  find  the  moment  of  inertia  of  any  body  turning  about  a  fixed 
axis,  with  great  accuracy,  no  matter  what  its  figure,  density,  or  the 
distribution  of  its  matter.  If  the  axis  do  nut  pass  through  its  centre 
of  gravity,  the  body  will,  when  dcfltctcd  from  its  position  of  equi- 
librium, oscillate,  and  become,  in  fact,  a  compound  pendulum  ;  and 
denoting  the  length  of  its  equivalent  simple  pendulum  by  Z,  we  have, 
after    multiplying  Equation  (378)  by  3£, 

J/. /.e  =  J/(/.v-  +  e2)  =  2mr2;     ....     (383) 


31  = 


W 


l.e 


9 

2  m  ?*2, 


(384) 


W 

~^  \\ 

in  which    W  denotes    the   weight  of  the   body.  '  i 

Knowing    the    latitude    of    the   place,  the   length    /'    of  the    simple 

second's  pendulum  is  known  from  Equation  (382)  ;    and  counting  the 

number    N   of    oscillations    performed    by    the    body    in    one    hour 

Equation  (379)    gives 

_  V '  (3G00)2 

To  find  the  value  of  e,  which  is 
the  distance  of  the  centre  of  gravity 
from  the  axis,  attach  a  spring  or 
other  balance  to  any  point  of  the 
body,  say  its  lower  end,  and  bring 
the  centre  of  gravity  to  a  horizontal 
plane  through  the  axis,  which  posi- 
tion will  be  indicated  by  the  max- 
imum reading  of  the  balance.  De- 
noting l>y  a,  the  distance  from  the  axis  C  to  the  i)o\ni  of  support  R, 

17 


25S 


ELEMENTS     OF     ANALYTICAL    MECHANICS, 


and   by    6,  the    maximum   indication    of  the   balance,    we   have,    from 
the   principle  of  moments, 


b  a 


We. 


Tlie  distance  a,  may  be  measured  by  a  scale  of  equal  parts.  Sub- 
stituting the  values  of  W,  e  and  I  in  the  expression  for  the  moment 
of  inertia,  Equation  (384),  we   get 


b.a.V .  (3CQ0)^ 


/. 


(385) 


If  the  axis  pass  through  the  centre  of  gravity,  as,  for  example, 
in  the  Jly-ivheel,  it  will  not  oscillate;  in  which  case,  take  Equation 
(383),  from  which  wc  have 


Mk;^  =  J/./.e 


J/e2. 


j\[ount  the  body  upon  a  parallel  axis  A,  not  passing  through  the  cen- 
tre of  gravity,  and  cause  it  to  vibrate 
for  an  hour  as  before ;  from  the  num 
ber  of  these  vibrations  and  the  length 
of  the  simple  second's  pendulum,  the 
value  of  I  may  be  found;  J/  is  known, 
b 'ing  the  weight  W  divided  by  g  ;  and 
e  may  be  found  by  direct  measure- 
ment, or  by  the  aid  of  the  spring 
balance,  as  already  indicated;  whence  k^  becomes  known. 


MOTION     OF   A   BODY   ABOUT   AN     AXIS   UNDEK    THE   ACTION     OF     IMPUL- 
SIVE  forces: 

g  237. — If  the  forces  be  impulsive,  we  may,  §  184,  replnce  in 
Equation  (30G)  the  second  differential  co-efficients  of  .r,  y,  z,  by  the 
first  differeutial  co-efficients  of  the  same  variables,  which  will  reduce 
it  to 


2  P  (  2  cos  a  —  X  C(3S  y)  —  I.  m 


zdx  —  xdz 
Tt 


=  0; 


MECHANICS     OF    SOLIDS. 


259 


and    replacing    dx   and   dz,    by  their  values    in    Equations    (3G8),  we 
find 

d-\j         1.  P  {z  cos  a  —  X  cos  y) 


dt 


(386) 


That  is,  the  angular  velocity  of  a  body  retained  by  a  fixed  axis,  and 
auhjected  to  the  simultaneous  action  of  imj^idsive  forces,  is  equal  to  the 
sum  of  the  moments  of  the  impressed  farces  divided  by  the  moment  of 
inertia  roith  reference  to  this  axis. 


,'1 


BALLISTIC    PENIJULUM. 

§  238. — In  artillery,  the  initial  velocity  of  projectiles  is  ascertained 
by  means  of  the  ballistic  pendulum, 
which  consists  of  a  mass  of  matter 
suspended  from  a  horizontal  axis 
in  the  shape  of  a  knifo-edge,  after 
the  manner  of  the  compound  pen- 
dulum. The  bob  is  either  made 
of  some  unelastic  substance,  as 
wood,  or  of  metal  provided  with 
a  large  cavity  filled  with  som^ 
soft  matter,  as  dirt,  which  re- 
ceives the  projectile  and  retains 
the  shape  impressed  upon  it  by  the 
blow 

Denote  by  F  raid  m-,  the  initial  velocity  and  mass  of  the  ball; 
F]  the  angular  velocity  of  the  ballistic  pendulum  the  instant  after 
ihe  blow,  /  and  M  its  moment  of  inertia  and  mass.  Also  let  / 
represent  the  distance  of  the  centre  of  oscillation  of  the  pendulum 
from  the  axis  xL  Tliat  no  motion  may  be  lost  by  the  resistance 
of  the  axis  arising  from  a  shock,  the  ball  must  be  received  in  the 
direction  of  a  line  passing  through  this  centre  and  perpendicular  to  the 
i.lane  of  the  axis  and  line   A  0.     With  thi>  condition.  Ivp  (380)  give? 

m-  V 
d^p       „        rn.V.l 

(It 


Fi  = 


VI  V 


^i,ir'         [M+  iJi)  I /.-;-  -f  e-) .       \JI-\-  inj.e 


260  ELEMENTS    OF    ANALYTICAL    MECHANICS. 

vihence 

^^      M  +  m         „ 

m 

aud  supposing  tliu  angular  velocity  coniinniiicated  to  the  penduinin  to 
be  equal  to  that  acquired  by  f;dliiig  from  i-e>t  through  tlie  initial  aic 
a,  in  Equation  (372),  we  have,  from  that  equation  and  Equation  (-iO), 
bv  wiitiiig  e  for  c?, 


r,=:  i/  'ly.         ^      ■  .  (1  _  cos  ,.)  =2  1/  .- -V— „•  SUl  \  a, 

r       "    [Jl  -f-  7/1}  [/c/  -f-  f.'-j  r     k-  -f-  e- 

and  Eq.   (374), 

t  /.■;  +  e' 

v,']tich  substituted  above  gives 

F,  rr  2  ^  •  sin  i  a  ; 

and  this  in  the  value  for  F  gives,  nfier  sub-tituting  for  the  ratio  of  the 
masses  that  of  their  weights, 

W  +  ?t.     c            .      ,  ,        , 

F=2 e  •  sm  .  a (387). 

From  thi>^  equation  we  mny  liiid  the  initial  velocity  F;  and  Vor 
this  piirp;ise,  it  will  only  be  necessary  to  have  the  duration  of  a  single 
o>cillation,  and  tlie  aiu[iliiude  of  the  arc  described  by  the  centre  of 
gravity  of  the  pendulum.  The  process  for  linding  the  time  has  been 
explained.  To  find  the  arc,  it  will  be  sufficient  to  atiarh  to  the 
lower  extremity  of  the  pendnlum  a  p'linter,  and  to  fix,  on  a  permanent 
si  and  below,  a  circular  graduated  groove,  whose  centre  of  curvature  is 
at  A]  the  grc;ove  being  tilled  with  some  soft  substance,  as  tallow,  the 
pointer  will  mark  on  it  the  extent  of  the  oscillation.  Knowing  thus 
t!ie  arc  «,  and  the  value  of  e,  found  as  already  described,  §  236,  wo 
have  V. 


MECHANICS     OF    SOLIDS.  261 


THE     GUN     PENDULUM, 

This  consists  of  ;i  gun  suspended  I'njui  ;i  lioiizoiilal  axis.  TIk-  >]iot 
is  tired  from  the  gun,  and  its  velocitv  is  inferred  from  tlie  recoil,  as 
in  the  Ballistic  Pendiiluiii.  The  forces  measured  by  the  quanliiii^s  of 
motion  developed  by  the  e.\p;insive  aetiuu  ot  the  exploded  powder, 
must  be  in  equilibrio.     Make 

V  =  velocity  of  the  ball  on  leaving  the  gun, 
nV  z=  average  velocity  of  the  inflamed  powder, 

F  =  angular  velocity  of  pendulum  on  parting  from  shot, 
W^  =  weight  of  gun  pendulum, 
W^  =         "  ball   and   wad, 

^V^  =        "  the  cliai'ge  of  powder  and  bag, 

W^,  =        "  "  "         of  powder  alone, 

6  =  diameter  of  bore, 

d  =  diameter  of  ball, 

e  =  distance  of  axis  of  bore  from  axis  of  suspension. 

The  quantit}^  of    motion    in    ball  and  wad,  on  leaving   the  gun,  will    be 

W,, 

V :    tlie    correspouiling    pressure    on    the    bottom    of   the    mm    is   to 

th;it  which  generates  this  motion,  as  the  area  of  a  cross-section  of  the 
bore  is  to  that  of  a  great  circle  of  the  ball.  Again,  the  blast  of  ihe 
powder  will  continue  its  action  on  the  gun  after  the  ball  leaves  it. 
Let  this  action  be  proportional  to  the  charge  of  powder.  The  moment 
of  the  force  impressed  upon  the  pendulum,  in  reference  to  the  axis  of 
suspension,  will  be  given  by  Eqs.  '(384)  and  (229)  ;  and  taking  the 
moments  of  the  other  forces  in  reference  to  the  same  axis,  we  have 

W  W,  X',       W  W 

—E.  v.  .l.e ■¥',-% '  .n-V-t ?.  jt'  •  F-  £  =  0  ; 

9  9  d-         g  g 

in  which    /<',  like  ??,  is    a    constant   to    be    determined    by    experiment; 

and  from  Avhich  we  find 

W  .v.. l.e 
V=- 


Wyc-  -+  "IF  .£+  n'W^.e 


2o2  ELE.UENTS     OF     A  :>:  A  L  Y  T  1  C  A  L     MECHANICS. 

TliL'    living   force    with    which    the    penduhim    separates   from    tlie    ball 

must   equal  twice  the  work  peifovmed  by  the   weight  while  the   centre 

of  gravity  is  moving  to  the  highest  point;    whence 

W 

v.- -l  •€  ~  2W  .  e  .  versine  a  =  4  TF  .  e  .  sin=  ^  a, 

9  ^  " 

ill    which    a    denotes    the    greatest    inclination    of    e    to    the    vertical. 

Whence 

F,  =  2^/i-smia; 
which  substituted  above  gives, 

W.-y^  +  nW  +nW 


V= ^, sin^a       ....     (388). 


The  methods  for  finding  e  and  a  are  the  same  as  in  the  ballistic 
pendulum.  To  find  n  and  n',  fire  the  ball  from  the  gun  into  the 
ballistic  pendulum;  the  effect  upon  the  latter  Avill  give  the  initial 
velocity  V.  Repeat  as  often  as  may  be  thought  desirable,  and  wiih 
different  charges.  The  corresponding  initial  velocities  substituted  in 
Eq.  (388),  will  give  as  many  equations  as  trials.  These  equations  will 
contain  only  n  and  n'  as  unknown  quantities,  which  may  be  found 
by  the  method  of  least  squares.  For  full  and  valuable  information 
on  this  subject,  consult  Mordecai's  "Experiments  on  Gunpowder." 


PAET    II 


MECHANICS    OF    FLUIDS. 


INTRODUCTORY     REMARKS. 

§239. — The  physical  condition  of  every  body  depends  upon  the 
relation  subsisting  among  its  molecular  forces.  When  the  attrac- 
tions prevail  greatly  over  the  repulsions,  the  particles  are  held  firmly 
together,  and  the  body  is  solid.  In  proportion  as  the  ditlerenco  be- 
tween these  two  sets  of  forces  becomes  less,  the  body  is  softer,  and 
its  figure  yields  more  readily  to  external  pressure.  When  these 
forces  are  equal,  the  particles  will  yield  to  the  slightest  force,  the 
body  will,  under  the  action  of  its  own  weight,  and  the  resistance 
of  the  sides  of  a  vessel  into  whicli  it  is  placed,  readily  take  the 
figure  of  the  latter,  and  is  liquid.  Finally,  when  the  repulsive  ex- 
ceed the  attractive  forces,  the  elements  of  the  body  tend  to  separate 
from  each  other,  and  require  either  the  application  of  some  extra- 
neous force  or  to  be  confined  in  a  closed  vessel  to  keep  them 
together ;  the  body  is  then  a  r/as.  In  the  vast  range  of  relation 
among  the  molecular  forces,  from  that  which  distinguishes  a  solid  to 
that  which  determines  a  gas  or  vapor,  bodies  are  found  in  all  possible 
conditions — solids  run  imperceptibly  into  liquids,  and  liquids  into 
gases.  Hence  all  classification  of  bodies  founded  on  their  physical 
properties    alone,  must,  of  necessity,  be  arbitrary. 

§  240. — Any    body    ^^■hose    elementary    particles    admit    of    motion 


264  ELEMENTS     OF     AXALYTICAL     MECHANICS. 

among  each  other,  is  called  a  Jluid — such  as  water,  wine,  mercury, 
the  :Mr,  and,  in  general,  liquids  and  gases;  all  of  which  are  distin- 
guishcd  from  solids  by  the  great  mobility  of  their  particles  among 
themselves,  This  distinguishing  property  exists  in  different  degrees 
in  different  liquids— it  is  greatest  in  the  ethers  and  alcohol;  it  is 
less  in  water  and  wine ;  it  is  still  less  in  the  oils,  the  sirups, 
greases,  and  melted  metals,  that  flow  with  difficulty,  and  rope  v.'hen 
poured  into  the  air.  Such  fluids  are  said  to  be  viscous,  or  to  possess 
viscosity.  T^inally,  a  body  may  approach  so  closely  both  a  solid  and 
liquid,  as  to  make  it  difticult  to  assign  it  a  place  among  either 
class,  as  paste,  putty,  and    the   like. 

8  241. — Fluids  are  divided  in  mechanics  into  two  classes,  viz.: 
compressible  and  inconqnessihle.  The  term  incompressible  cannot,  in 
strictness  of  propriety,  be  applied  to  any  body  in  nature,  all  being 
more  or  less  compressible ;  but  the  enormous  power  required  to 
chano-e,  in  any  sensible  degree,  the  volumes  of  liquids,  seems  to 
justify  the  term,  when  applied  to  them  in  a  restricted  sense.  The 
gases  ai'e  highly  compressible.  All  liquids  will,  therefore,  be  regarded 
as    incompressible ;    the  gases  as    compressible. 

8  242. — The  most  important  and  remarkable  of  the  gaseous  bodies 
is  the  atmosphere.  It  envelops  the  entire  earth,  reaches  far  beyond 
the  tops  of  our  highest  mountains,  and  pervades  every  depth  from 
which  it  is  not  excluded  by  the  presence  of  solids  or  liquids.  It 
is  even  found  in  the  pores  of  these  latter  bodies.  It  plays  a  most 
important  part  in  all  natural  phenomena,  and  is  ever  at  woyk  to 
influence  the  motions  within  it.  it  is  essentially  composed  of  oxygen 
and  nitrogen,  in  a  state  of  mechanical  mixture.  The  foimer  is  a 
supporter  of  combustion,  and,  with  the  various  forms  of  carbon,  is 
one  of  the  T^rincipal  agents  employed  in  the  development  of  mechan 
ical  power. 

The  existence  of  gases  is  proved  by  a  multitude  of  facts.  Con- 
tained in  an  inflexible  and  impermeable  envelope,  they  resist  pressure 
like  solid  lodies.  Gas,  in  an  inverted  glass  vessel  plunged  into 
water,  will  not  yield  its  place  to  the  liquid,  unless  some  avenue  of 
escape    be    provided  for    it.     Tornadoes  which   uproot    trees,  overt iiin 


MECHANICS     OF     FLUIDS.  265 

houses,  and  devastate  entire  districts,  are  but  air  iu  motion.  Air 
opposes,  I)}'  its  inertia,  the  motion  of  other  Lodies  through  it,  and 
this  ojjposition  is  called  its  resistance.  Finally,  we  know  that  wind 
is  employed  as  a  motor  to  turn  mills  and  to  give  motion  to  ships 
of  the  largest   kind. 

§  243. — La  the  discussions  which  are  to  follow,  fluids  will  be  con- 
sidered as  without  viscosity  ;  that  is  to  say,  the  particles  will  be 
supposed  to  have  the  utmost  freedom  of  motion  among  each  (ith;r. 
Such  fluids  are  said  to  be  'perfect.  The  results  deduced  upun  liie 
hypothesis  of  perfect  fluidity  will,  of  coursf,  require  modiiication 
when  applied  to  fluids  possessing  sensiljle  viscosity.  The  nature  and 
extent  of  these  modifications   can   be    known    only  from  experiments. 


makiotte's  law. 

§244. — Gases  readily  contract  into  smaller  volumes  when  pressed 
externally  ;  they  as  readily  expand  and  regain  their  former  dimen- 
sions .when  the  pressure  is  removed.  They  are  therefore  both  com- 
jyi-cf^ihle  and  elastic. 

It  is  found  by  experiment,  that  the  change  in  volume  is,  foi-  a 
constant  temperature,  always  directly  proportional  to  the  change  of 
pressure.  The  density  of  the  same  body  is  inversely  proportional  to 
the  volume  it  occupies.  If,  therefore,  P  denote  the  pressure  ur;on 
a  unit  of  sui-flice  which  wiil  produce,  at  a  given  temperature,  say 
0°  Ceutr.,  a  density  equal  to  unity,  and  D  any  other  density,  and 
n  the  pressure  upon  a  unit  of  surtiice  which  will,  at  the  same  tem- 
perature of  the  gas,  produce  this  density,  then,  according  to  the  ex- 
periments above    referred    to.  will 

p  =-.  P.D (389) 

This  law  was  investigated  by  Boyle  and  Mariotte,  and  is  known 
as  Mariotte^s  Law.  By  experiments  made  at  Paris,  it  was  found  that 
this  law  obtains,  Avhen  air,  in  its  ordinary  condition,  is  condensed  27 
and  rarefied    112    times. 


266         ELEMENTS    OF    ANALYTICAL    MECHANICS. 
LAW     OF     THE     PRESSUKE,     DENSITY,      A^'D     TEMPEEATCEE. 

§  245. — Under    a   constant  2'»'^ssure,    all     bodies    are    expanded    by 

heat ;    under  a  constant  volume,  their  elastic  forcf  is   increased  by   the 

same    agent.       Experiment  has    shown  that  the  laws  of  these  chanjies 

for   gases    are    expressed    by  'P  ^    /u 

;>  =  P./).(1  +«^);     •     -^^4-^  ^(390) 

in    which    p    denotes    the     pressure     upon    a    unit    of    surface,    D    the 

density    of    the    gas,    ^    the    difference     between    the    actual    and    some 

standard   temperature,  and  a  a  constant  which  is  equal  to  —g-  — 0.003CG5 

when  the  standard  is  0°  centr,,   and  &  is  expresst'd  in  units  of  that  scale. 

First  supposing   D   and    &   variable   and   ^j  constant;     then  jt>  and  ^ 

variable    and    D   constant,    Equation    (-jOO)  gives 

d  D  a .  I>  dp  a.p 

U  ~  ~  1  -I-  ad  '       Vd  ~  YT'aJ      •      •      •     •       (•1) 

The  quantity  of  heat,  denoted  by  q.  uicessary  to  change  the  tem- 
perature ^  degrees  from  the  ass-nmed  standard,  will  be  a  function 
of  p,  D,  (5  ;     b'ut    because    of  Equation    (SDO.)    we    may    write 

q=;f{D,p)  .  .  .  .  .(b) 

The  increment  of  heat  which  will  raise  a  body's  temperature  one 
degree,  is  called  its  s]jecijic  In  at.  The  s)  ecific  heat  being  the  in- 
crement of  q  for  each  unit  of  d.  if  c  denote  the  specific  heat  when 
the    pressure    is    constant,   and    c^   that    when    the    density    is   constant, 

then  will 

dq        dq     d  D  dq        dq    dp 

^^Td~TD"dl''       '^ '  ^  Jd' ~  Tp'Tb' 

or,    Equations    (a), 

d  q       u. .  D  d  q       a.p 

'^  ^  ~  (TI*  *  rT«^  '       '^'~  d~p'Y'-^ir&'' 

and   by    division,    making    czzzy.c^,  C   *^' 

^     dq  d  q 

a  1)  d  p 

in  which  7,  denotes  tiie  ratio  of  the  specific  heat  of  the  gas  at  a 
constant  pressure  to  that  at  a  C(»nstant  density.  This  ratio  is 
known  from  experiment  to  be  constant  f  .r  atmospheric  air,  and  is 
probably     so     for    all     gases.        'J  he     ixjiii  imcnls    of    Desonius     and 


MECH/VNICS    OF    FLUIDS.  267 

Clements  make  its  value  l.'>482-,  tliose  of  Gay-Lussac  and  ^Vajl-.r 
l,o748;  and  those  of  Dulung  on  pertectly  dry  air  1,421.  Regard- 
ing  y    as    constant,    tha    integration    of    the    foregoing    equation    gives 

1 


-f(9 


r-. 

(See  Appendix  No.  3.) 


in  which  /,  denotes  any  arbiti-ary  lunction  of  the  quantity  within 
the  jjarenthesis,  and  from  which,  denoting  the  inverse  functions  by 
i^,    we   may    write 

p=.l/.F{q) (c) 

From    Equation    (390),    we    have  ^.r-y-t   — "'"  r-' 

Sudden  compression  increase.^,  and  a  sudden  expansion  decreases  the. 
temperature^  of  bodies,  and  if  q  remain  the  same,  while  suddenly 
p,  I),  6,   become  p\D\6',    we    have  t>  - 

p'^D'^.F{cj),    .     .     (e)  &'^-l^D'r-Kjr^rj)-l     .     .     (g) 


Eliminating  F  (q)  first  from  Equations  (c)  and  (e),  and  then  fiom  Equa- 
tions (d)  and  (g),  we  have,  replacing  7  and  a  hy  their  numerical  values, 

.2>'\  1,421  /v-''  /V   .^>«f 

P'=p{jj)  •         •     -"'         ■       (391) 

e'=  (273  +  ^)  {-jj)  -  273      .         .         .      .  (392) 

These    iquntiiins    give    the    relation    between    the    densities,   elastic 
fi  ices,    and    the    teiirj  eralures  of  a    gas   tncldtnh    compresstd    or  dila- 
ted,   and    retaining    the    quantity    of   its    heat    uiichaiiged.  /t  -^   f^/J,  r 
The  pressure  being  coiistant,  maUe,in   Eq.  (390),  t)  =  0,  Z)  =  D^,  and   /  ,  ^ 
divide  same  equation  by  the  result  ;   we  find  JJ  =  D^  -^  (1  -\-a6).       Make  /• 
p=zlJ,„-  h^^  •  </'  =  weight  of  a  column  of  mercury  at  standard  tempiM'atuie'JO-    /C 
T.  and  resting   on  a  base  iniity,  in  Lat.  45°,  where  gravity  is  (f ,     These 
in  Eq.  (389)  give,  after  writing  0,0i  204  Ji.r  a,  and  ^°  —  32°  for  ^, 

P  -  :?"-A_i^  .  [1  4  (^o  _  g2°) .  0.00204]  •     .     .     {\.\^'X) 
If    the    tcii'i  i^iati  If    of   ll*-    n:<riur\      v;.r\     fj.m    ihe    .>-tai  d.M'.l    T, 


2G8 


ELEMENTS     OF     ANALYTICAL     MECHANICS. 


and  become    T'  then   will   B,,^  aL-o   varv   and    lieconie   D[^.  and  ti»  exert 
the    same  pressure    li^^    must    liave    a    new    lieight    /(,    and    such  that 

D,,.h,^.g'  =  D',„.h.y'. 
Mercury    expands    or    'odiitracts    O.OOOlOOP*'     pait    of    its    entire    vol- 
ume   for    each    degree   vi  Fain,    by    which    it   increases   or    diminishes 
its    temperature.        iVnd     as    the    density    of    the    same    body    varies 
inversely    as    its    volume,    we    have 

i>:  =  D,„  [1  +  {T-  r)  -0,0001001] 
which    suiistituted    above    gives 

^,,  =  A  [1  +  (T'-  r).  0,0001001] (3<J4) 

EQUAL     TEAX8MISSI0N     OF    PEESSUIIE. 


§246. — Let  E H  L,  represent  a  closed  vessel  of  any  shape,  with 
which  two  piston  tubes  A  B'  and 
D  C  communicate,  each  tube  be- 
ing provided  with  a  piston  that 
fits  it  accurately  and  which  may 
move  wit*liin  it  with  the  utmost 
freedom.  The  vessel  being  filled 
with  any  fluid,  let  forces  P  and 
P',  be  applied,  the  former  per- 
pendicularly to  the  piston  A  B, 
and  the  latter  in  like  direction 
to  the  piston  CD,  and  suppose 
these    forces    in    equilibrio,    which 

thry  may  be,  since  the  fluid  cannot  escape.  Now  let  the  piston 
A  1)  be  moved  to  the  position  A'  B' \  the  piston  CD  will  take 
so  Me  new  position,  as  CD'.  And  denoting  by  s  and  *•',  the  dis- 
lances  .1  J'  and  C  6",  respectively,  we  have,  from  the  principle  of 
virtual   velocities, 

P  H    =    P'  h'. 


Denote  the  area  of  the  piston  A  B  by  o,  and  that  of  the  piston 
CD  by  a',  then  will  the  volume  of  the  fluid  whicli  was  thrust  from 
the    tube  A  B\  be  measured  by  a  .  s,  and  that  which  entered  the  tuT)e 


MECHANICS     OF    FLUIDS  209 

t>  C,  will  be  measured  hj  a'  s'.  But  the  pressuie  upon  the  pistons 
and  the  temperature  remaining  the  same,  the  entire  volume  of  the 
fluid    in    the    vessel    and    tubes    will    be    unehansred.       Hence, 


dividing  the  equation  above  by  this  one,  we    have 

P         P' 

-  =  ^ (39G) 

a  a 

Tliat  is  to  say,  two  forces  ai->'plied  to  jiistons  lohich  communicate  freely 
with  each  other  through  the  intervention  of  some  confined  fluid,  zvill 
be  in  equilihrio  when  their  intensities  are  directly  proi^ortlonal  to  the 
areas    of  the  instons  upon  which  they  act. 

This  result  is  wholly  independent  of  the  relative  dimensions  and 
positions  of  the  pistons ;  and  hence  we  conclude  that  any  jiressiire 
co)n7nvnicated  to  one  or  more  elements  of  a  fluid  m((ss  in  equilihrio,  is 
equally  transmitted  throughout  the  whole  fluid  in  every  direction.  This 
law  which  is  fully  confirmed  by  experiment,  is  known  as  the  prin- 
ciple of  equal  transmission  of  pressure. 

§  247. — Let  a  become  the  superficial  unit,  say  a  square  inch  or 
square  foot,  then  will  P  be  the  pressure  applied  to  a  unit  of  sur- 
face, and.  Equation   (390), 

P'  =  Pa'. (807) 

«.  That  is,  the  pressure  transmitted  to  any  portion  of  the  surface  of 
the  containing  vessel,  will  be  equal  to  that  applied  to  the  unit  of 
surface  multiplied  by  the  area  of  the  surface  to  which  the  trar.smis- 
sion    is   made. 

§248 — Since  the  elements  of  the  fluid  are  supposed  in  equi!il)rio, 
the  pressure  transmitted  to  the  surface  through  the  elements  in  con- 
tact with  it,  must,  §  217  and  Equations  (332),  be  normal  to  the  sur 
face.  That  is,  the  pressure  of  a  fluid  against  any  surface,  acts  always 
in    the    dlredlon    (f  the    normal. 


270 


ELEMENTS     OF     ANALYTICAL     MECHANICS. 


^^ 


MOTION    OF   THE   FLUID    PARTICLES. 

§  249. — The  particles  of  a  fluid  having  the  utmost  freedom  of 
motion  among  one  another,  all  the  forces  applied  at  each  particle 
must  be  in  equilibrio.  Regarding  the  general  Equation  (40)  as  ap- 
plicable to  a  single  particle,  whose  co-ordinates  are  x,  y,  z,  we  shall 
have 

a;  =z  a-, ,     y  =  y^,     s  =  z, , 

and   suj^posing  the   particle   to    have   simply  a   motion  of  translation, 
we  also  have 

^9  =  0;     5-^  =  0;     ^^  =  0; 

and   that   equation   becomes 

cPx- 


ylP  cos  «,  —  m  •  -— ;-  \  ^  X 

+    (2PCOS/3  -  m.-^)  ^y     }^   =  0 

/  (Pz\   ^ 

-f     (2  i^cos  y  —  m  •  -yrr  )  ^  ' 


whence,  upon    the    principle  of  indeterminate    co-effici'-nts, 

(/^  X 


2  P  cos  a 


d  fi 


0; 


2  P  cos  /3  —  m  ■  -~  —  0 

cPz 
2  P  cos  7  -  «i .  -^  =  0. 


(39S) 


Now  the  terms  2  P  cos  a,  2  P  cos  /3  and  2  P  cos  7,  arc  each  composed 
of  two  distinct  parts,  viz.  :  1st.,  the  component  of  the  resultant  of 
the  forces  applied  directly  to  the  particle ;  and  2d.,  the  component 
of  the  pressure  transmitted  to  it  fi'om  a  distance,  arising  from  the 
forces  impressed   upon   other   particles. 

Denote  by  X,   Y  and  Z,  the  accelerations,  in  the  directions  of  the 
a.xes   X,  y,  z,    respectively,    due  to   the  forces   applied  directly  to    tiie 


MECHA.NICS     OF     FLUIDS. 


271 


particle ;  then  ???,  being  the  mass   of  the  particle,  the   components   of 
the  forces  directly  impressed  will  be 

m  X  \     m  Y;     mZ. 

The  pressure  transmitted  will  depend  upon  the  particle's  place, 
and  will  be  a  function  of  its  co-ordinates  of  position.  Denote  by  |^, 
the  pressure  upon  a  unit  of  surface,  on  the  supposition  that  every 
point  of  the  unit  sustains  a  pressure  equal  to  that  communicated  to 
the    particle  from  a   distance ;    then  will 

p  =  i^(.r,y,z). 

Conceive  each  particle  of  the  fluid  to  consist  of  a  small  rectan- 
gular  parallelopipedon  whose 
faces  are  parallel  to  the  co- 
ordinate planes,  and  whose  con- 
tiguous edges  at  the  time  t^ 
are  cLr,  dy  and  dz\  and  let 
X,  y,  2,  be  the  co-ordinates  of 
the  molecule  in  the  solid  an- 
gle nearest  the  origin  of  co- 
ordinates. Then  would  the 
ditference  of  pressure  on  the 
npposite  fiices,  which  are  j)aral- 
lel    to   the   plane   zy,  were    these  fiices    equal    to    unity,  be 

dp 


/ 


V 


/Y 


F  [x  -I-  dx,  y,  z,)  —  F  (.r,  y,  z,)  =-^  •  dx. 

and    upon  the   actual    faces    w'hose    dimensions    are   each    dz.dy,  this 
difference    becomes.  Equation  (397), 

dp 


d  X 


dx-dy  •  dz. 


In   like    manner    will    the    difference    of   the    pressures   transmitted 
Jo    the    opposite  faces  parallel  to    the    planes  zx    and    xy^  be,  respec- 
tively, 

-~-  ■  d  y  •  d z  •  d X.     and     —r-  '  dz  '  d x  •  dy. 
dy        -^  '  dz  ^ 


■^72     ELEMENTS  OF  ANALYTICAL  MECHANICS. 


These  pressures  being  normal  to  the  surfaces  to  which  they  are 
respectively  applied,  they  will  act,  the  first  hi  the  direction  of  x, 
the  second  in  the  direction  of  y,  and  the  third  in  the  .direction 
of  z.  And  as  these  differences  alone  determine  that  portion  of  the 
motion    due  to    the   transmitted    pressures,  we    have 


2  P  cos  a  =  711 X 


d'p 
d  X 

d]) 


d X . d  y . dz 


2  P  cos  /3  —viY f-  •  dy  .  dx  .dz; 

dy 


2  P  cos  y  =z  m  Z 


-——  •  dz  .  d X  .  d y. 

dz  -^ 


Denote  by  D  the  density  of  the   mass  ?»,  then  wall,  Equation  (1)'. 
rii  =  D  .  d  X  .  II  y  .  d  z, 
and   by  substitution,  Ei[uations  (?)98)  become 


1 

dr 

I) 

dx 

1 

d'p 

1) 

dy 

1 

dp 
dz 

^  ^  ~  ~dl? 

dfi 


(399) 


A    Denote   by    u,  v  and  w,  the   velocities    of  the    molecule  whose  co^ 
/    ordinatcs    arc    xyz,  parallel   to   the   axes    x,  ?/,  z,  respectively,  at    the 
time  t.     Each    of  these  will    be    a    function    of  the   time    and   the   co- 
ordinates   of  the    molecule's  place;  and,  reciprocally,  each  co-ordinate 
will  be  a  function  of  t,  u,  v  and  tv  ;  whence.  Equations  (12)  and  (13), 


(Px 
17' 


lu\     dt    ,    dii    dx    I    du      dy    ^   du     dz 


X        du        /duX     dt       du    dx       du     dy    ^   du 
'^  ~  dt  ^  ^d  t/      dt       d  X    d  t        dy      d  t       dz 


dx  dy  dz      ^        ,    .         ,  .•     T 

and  rcnlacino;  — 5  —^5  -^5    by  their  values  u,  v,  to,  respectiveiv,  we 

''  dt  dt  at 

have 


d"^  X 


/  du\ 
\~dJ/ 


du  du  du 

-J—  ■  u  +  -—  .  V  +  -7—  .  w ; 
dx  dy  dz 


MECHANICS     OF     FLUIDS, 


27. 


in    the   same  way, 


dv 


d  t"  \  d  t  y  dx  dy 

d'^z         /  dtu\         dw  dw 

d  P  \  d  t  y  d  X  d  ij 

which,  substituted  in   Equations  (399),  give 

--(4^) 
--(4^) 

1       dp  /dw\         d  IV 

D  '  dz    ~        ~   \'dJ/    ~  dx 


1       dp 

U'  ITx 

i-    'In 

'D'~d7j 


dto 
dx 

dv 
dx 

d  IV 


d  u 

dy 

d  V 
dy 

d  20 

dy 


dv 

V  +    -y-  ■  W, 

dz 


dto 

V  -\ w 

dz 


du 

dv 
'--dj""'^ 

dto 

V   — •  20. 

dz 


(400) 


Here  are  three  equations  involving  five  unknown  quantities,  viz.  • 
w,  V,  w,  p  and  -D,  which   are   to   be  found  in   terms  of  .r,  y,  z  and  t. 

Two  other  equations  may  bo  found  from  these  considerations,  viz  : 
the  velocity  in  the  direction  of  .r,  of  the  molecule  whose  co-ordinates 
are  x  y  z,  is  u ;  the  velocity  of  the  molecule  in  the  angle  of  the 
parallelopipedon  at  the  opposite  end  of  the  side  d  x,  at  the  time  t,  is 

u  -j — ~  'd  x: 
dx         ' 

and   hence   the   relative   velocity   of  the   two  molecules   is 


du 

u  +  -; —  dx 

dx 


du 

u  =  - —  d  X. 

dx 


At  the  time  t,  the  length  of  the  edge  joining  these  molecules  is 
dx^i  and    at   the    end    of  the   time   t  ^  d  t^  this  length  will    be 

7       ,    <^ ^'     7        7  7     /  -.        du     ^  . 

dx  +  -T—  •  dx  .  dt  —  d  X  (I  +  -r-  •  dt); 
d  X  ^  dx 

the  second  term  being  the  distance  by  which  the  molecules  in 
question  approach  toward  or  recede  from  one  another  in  the 
time  dt. 

18 


274  ELEMENTS     OF     ANALYTICAL     MECHANICS. 

In  the   same  way  the  edges    of  the    parallelopipedoii  which   at    the 
time  t,  were  d  7/  and  d  z,  become  respectively, 

d  V      ^        7  7     /,     ,     "^^i'      7  N 

dy  ■\ — - —  dij  .at     =  dij  [I  -\ — - —  at); 

dto      ^        ,  J     ,-.     .     dw      ,  . 

dz  -\ —  ■  dz.dt     =  J  s:  (1  H —  -dt); 

d  z  d  z 

and   the   volume   of  the  parallelopipedon,  which   at   the   time    t^    was 
dx  .dy  .dz^  becomes  at   the   time  t  -\-  dt., 

,     ,  du      ,  ,     , ,        d  V      ,  ,     , ,         dw 

rf,.rf3,.rf,(,  +__.,„). (1 +„.,„). (1+_.,U). 

The   density,  which   was   7),  at   the    time   ?,  being  a  function  of  xyz 
and  t,  becomes   at   the    time   i  +  dt, 

dD     ,         dD     ^  dD     ^  dD     ^ 

D  -\-  -J-  ■  d  t  +  -—  ■  d  X  +  -^  ■  dy  + dz; 

d t  d X  dy  dz 

which   may    be   put    under   the    form, 

/dD         dD   dx         dD    dy         dD    dz\ 

y  d  t  dx     dt  dy     dt    -      d  z     d  t^        ' 

and  replacing 

d  X        dy         dz 
d t         dt         dt 

by  their  values  u,  v,  w,  respectively, 

/dl)         dD  dD  dD      \    , 

\  dt  d  X  d  y  dz       y 

Multiplying  this  by  the  volume  above,  we  have  for  the  mass  of    the 
parallelopipedon,  which  was 

D  .  d  X  ,  dy  .  dz, 

at  the   time   t,  the  value. 

r  /([D        dD_         ,(j^        ,d_D_      \       1 

L  \d  t  d  X  dy  dz         '        J 


dy 

dy 


at   the  time    t  -\-  d  t. 


MECHANICS     OF    FLUIDS.  275 

But  these  masses  must  be  equal,  since  the  quantity  of  matter 
is  unchanged.  Equating  them,  striking  out  the  common  factors,  per- 
forming the  multiplication,  and  neglecting  the  second  powers  of  the 
differentials,  we  have 


f  d%i.         dv        dw\        dD        dD  dD  dD 

\  d X  d  y         dz  /  dA         a  x  dii  d  z 


This  is  called  the  Equation  of  continuity  of  the  fluid.  It  expres- 
ses the  relation  between  the  velocity  of  the  molecules  and  the  den- 
sity of  the  fluid,  which  are  necessarily  dependent  upon  each  other. 
This  is  a  fourth  equation. 

§250. —If  the  fluid  be  compressible,  then  will  the  fifth  equation 
be   given    by  the    relation, 

F{D,p)  =  0, (402) 

as  is  illustrated  in  the  particular  instance  of  Mariotte's  law.  Equa- 
tion (389).  The  form  of  the  function  designated  by  the  letter  i^ 
will    depend    upon    the   nature    of  the    fluid. 

§251. — If  the  fluid  be  incompressible,  the  total  differential  of  Z) 
will    be   zero,  and 

dD  dD  dD  dD 

d t  dx  dy  d  z  ^ 

and  consequently,  the  equation  of  continuity,  Equation  (401),  becomes. 


r    -  -—  -f— -+^-=0; (404) 

I X  d  II  d.  z  ^        ' 


\f- '  ^  d  u  dv  dto 

1-  ■ —  H 

^  dx  dy  dz 

and   wc   have    for     the    determination    of  u^  v,  iv.  D    and   p,    the    five 
Equations  (400),  (403),  (404).  ^  ^/f 

g  252. — These  equations  admit  of  great  simpliffcation  in  the  case 
of  ai.  incompressible  homogeneous  fluid  when  U'dx-\-  v.dy  -f-  xo.dz^ 
is  a  perfect  differential.     For  if  we  make 

u  d  x  +  V  d.y  +  V)  d  z  =i  d  (p, 


276  ELEMENTS     OF    ANALYTICAL    MECHANICS, 

then    fi'om   the   partial   differentials  Avill 

dCD  d  CD  d(D  ,     „ 

''  =  7!'  "  =  77'  "^in-'  ■  ■  ■  ■  (^''^J 

which,  in  Equation  (404),  gives  for   the   equation    of  continuity, 
d'^  (Q         d"  (p        d"^  CD        ^ 

by    the   integration   of  which   the   function  9  may    be    found. 
Differentiating  the  values  of  ?<,  v  and  w  above,  we  have 

,  d"^  m  d"  CD  d'^  cp 

du  =  — —  ;    d  V  =  -——  :     d  lu  =  — —  • 
dx  d  If  ■  dz 

Eliminating  ?/,  v,  w,  d  u.  d  v  and  d  20,  from  Equation  (400),  by  means 
(;f  the    values  of  these  quantities  above,  we  have 

I      d  p  (/-  (p  d  cp     d^  cp         d  cp        (F  cp  d  cp        d^  9 

A 


Ij      dx  dx-dt         dx     d  x"^        dy     dx.dy        dz     dx.dz 

1      d  p  d'^  9  dcp        d'^  cp  d<p    d^  co        d  cp        d^  9 


D     dy  dy  .d  t         d  x     dy  .  d  x       d  y   d  y^        dz     dy  .dz  ' 

I      dp  d^  cp  dcp        d^  cp  dip       d'-^  cp  dcp     d^  cp 

D      dz  dz.dt         dx    dz.dx        dy     dz.dy        dz     dz"^ 

Multiplying  the  first  by  (/.r,  the  second  by  dy^  the  third  by  f/2,  and 
aiding,    we    iiad, 

l.,=x,.+  r..+^,.-4^-..[(f|)%  (1-;)%  (|i)](40.) 

From  wliich,  by  inti^gration,  may  be  found  the  pressure  at  any  point 
of  an  incompressible  fluid  mass  in  motion,  when  Equation  (400)  is 
the    equation  of  continuity. 

§253. — When  the  excursions  of  the  molecules  arc  small,  the 
second  powers  of  the  velocities  may  be  neglected,  which  will  reduce 
Equation   (407)  to 

^■dp  =  Xdx  +  Ydy  +  Zdz  -   d  '^-     •     •     (408) 


i[ E  C  H  A X  1  C  S     OF     FLUIDS.  2  ,  » 

^254. — If  the  condition  expressed  by  Equation  (400)  be  not  ful- 
filled, then  we  must  have  recourse  to  Equation  (404)  to  find  the 
pressure. 

§255. — Resuming  Ecj[uation  (401),  which  appertains  to  a  conipres- 
sible  fluid,  retaining    the    condition  that 

u  d X  -\-  V  dy  -\-  lod z  ^  d cp 

is    a   perfect  differential,  and  from  which,  therefore, 

u  =  ——;     V  z=  —— ;     76.  =  -—  ;      .      .     .     (409) 
dx  dy  dz 

we    obtain  by  substitution, 

^\dn  dv        dw  )        dD      d  D  d^         dDd(D        d  D  d(r> 

(    a X  d y        d  z    ^         d  t        d  x    d  x         d y    dy         a  z    a  z 

If  the  excursions  of  the  molecules  from  their  places  of  rest  be 
very  small,  both  the  change  of  density  and  velocity  will  be  so 
small  that  the  prod  nets  which  constitute  the  last  three  terms  of 
this  equation  may  be  neglected,  and  the  equation  of  continuity  be- 
comes 

^      /  (/  y/  d  V  d  10  \  d  D 

^•(77+  rf7+  rfr)+^  =  ''l 

and  replacing  d  ?<,  d  v  and  d  w,  by  their  values  from  Equations  (400). 
and  dividing  by  Z>,  we  find 


1: 


d\i)%I>    ,    d*^         d"-^         d'"(p 

^       ^''^-  -dT-  +  ^  +  .7F  "^  ^  ^  ^'    *    '    ■     ^    ^ 

from  which,  and  Eq.  (408),  the  equation  connecting  the  extraneous 
forces  with  the  co-ordinates  x  y  z,  and  that  expressive  of  Mariotte's 
law,  the  function  9  niay  be  found,  then  the  value  of  D,  and  finally 
that  of  p. 

The    excursions   being    small,    if  we    impose    the    additional    condi- 
tion  that   the    molecules  of   the    fluid    are    not    acted    upon   by    extra- 


27S  ELEMENTS     OF     ANALYTICAL    MECHANICS. 

nev^L..'5  forces,  in   which    case    the    motions    can    only  arise   from    some 
arl)itrary   initial   disturbance  ;    then,  Equation  (408), 

1  7      f^9  C?^23 

and    by  ISIariotte's   law, 

p^P.D  =  a'-.D (411) 

from  which 

diy  =  cC-dD (412) 

and  the  above  may  be  written,  after  dividing  by  d  /, 

a?-    dD  _    ,    d\ogD  _       d^cp  (^.^■^\ 

dt'1)   ^"'''      d^t      ~~  Jf ^    '' 

wliich,  in  Equation  (410),  gives 

dt^  -"   \dx^   ^  rfr         dz-^J  ^^^^^ 

From  this  Ecjuation  the  function  9  is  to  be  determined,  then  the 
value  of  Z>,  from  Equation  (410),  and  that  of  p.  from  either  of  the 
Equations  (411)  or   (413). 

§  256. — Let  r  be  a  function  of  ip.r//2-,  and  such  that 

'!■'  !?_d'o         d'  c         cP_0 
d  f^        d  x'        d  if-       d  '/ 

a  condition  obviously  fulfilled.  Equation  (-H4),  when 

r  =  cr  1;  =1=  r, 

and  in  wliich  c  is  any  arbitrary  constant.    Dividing  Equation  (414)  by  that 
above,  we  find 

To  integrate  this,  add  to  both  members 

(/•  o 
dr  .dt^ 


MECHANICS     OF     FLUIDS.  279 

and    we   shall    have 

1       ,  /f^<P      ,  d(p\         a         /dcp  d(p\ 


and   making 


we   have 


4f +  <.-4^  =  F, 

at  a  r 


dV   _        d V  ^ 
~dt    ~      '   dr   ^ 

and   V  being   a  function  of  x  and  t^  we  have,  by  difterentiating; 

,^       dV      ,  dV     ^ 

dV  —  — —  ■  dt  -{-  -—  'dr  ' 
dt  dr  ' 

or  by  substituting  for    — ; —   its  value   above, 
^  °  dt  ' 

(IV  ,  ^  ,  ,         f?F     ,,      , 

dV  =  -—  id  r  +  adt)  —  -7-  •  (Z  (r  +  a  0. 

and  by  integration, 

tZ  f  dr  ^  ' 

in  which  F'  denotes    any   arbitrary  function. 
In  like    manner,  by  subti'acting 

d"^  ^5^ 

a  • —» 

d  t  •  d  r 

from    both   members  of  Equation  (415),  we  find 

dm  dcp  .,  ,  , 

in  which  /'   denotes   any  arbitrary  function. 
Whence,  by  addition, 

'^  =  \F'{r^at)  ^y'{,-~at\ 


2S0 


ELEMENTS     OF     ANALYTICAL     MECHANICS. 


and   by  subtraction, 
But 


whence. 


l.iP^  (.  +  «,)_!/'    {r-ai). 


7  '^^'P  7  >  <^^®  7 

a  CO  =^  — ; —  •at  A r-^  •  a  r 

^         dt  dr 


dcp  =  -—  •  F'  {r  -[-  at)d  [r  -\-  at)  —  -^—  •  /'  (?•  —  a  t)  d  (/•  —  at)\ 


and   by  integration, 


(p  =  Fi^r  -\-  at)  +/(?•  —  at) 


(416) 


in   which  F  and  /,  denote   any   arbitrary   functions  Avhatcver,  and  are 
determined   from    the   initial    conditions   of  the   question. 

This  Last  formula  is  used  in  discussing  the  subject  of  sound,  and 
the  more  general  equations  which  go  before  are  employed  in  devel- 
oping the  principles  of  light  and  heat  as  well  as  those  of  the  tidal 
waves  of  the  ocean   and   of  the  atmosphere. 


EQUILIBKIOI    OF    FLUffiS. 


§  257. — If  the   fluid  be   at   rest,  then  will 
d^  X  d^  y  d'^  z 


and   Equations  (399)  become 

dji 
d  X 

dp 
dy 

dp 
d  z 


=  D.X;^ 


=  D.Z. 


(417) 


§258. — Multiplying  the  first  by  dx,  the  second    by  d  y,  the  third 
by   d  Zy  and  adding  we  find, 

d2)  =  D  {Xdx  +  Ydy  -)-  Zdz);  •     .     .  •  .    (418) 


MECHANICS     OF    FLUIDS.  281 

and  by  integration, 

p  -  fn  .{Xdx  +   Ydy  +  Zdz);    •     •     .     .(419) 

whence,  in  order  that  the  value  of  j9  may  be  possible  for  any 
point  of  the  fluid  mass,  the  product  of  the  density  by  the  function 
X dx  -f-  Ydy  -\-  Zdz,  must  be  an  exact  differential  of  a  function  of 
the  three  independent  variables  x,y,z.  Reciprocally,  Avhen  this  condi- 
tion is  fulfilled,  not  only  will  the  pressure  at  any  point  become  known 
by  substituting  its  co-ordinates,  but  the  Equations  (417),  will  be  s.it- 
isfied,  and  the  fluid  will  be  in  equilibrio. 

§  259. — Conceiving  those  points  of  the  fluid  which  experience  equal 
pressures  to  be  connected  by,  indeed  to  form  a  surface,  then  in 
passing  from  one  point  to  another  of  this  surface,  we  shall  have 
dp  =  0,  and 

Xdx  +   Ydy  -f  Zdz  =  0, (420) 

which    is   obviously    the    ditferential   ecjuation    of  the    surface. 

Dividing  this  b}'  Rds,  in  which  ?>iiA',  denotes  the  resultant  <>f  the 
forces  which  act  upon  any  particle,  and  da,  the  element  of  any 
curve  upon  the    surface  passing  thi'ough    the    particle,  we  have 

Xdx  Y    dy_    .     Z     d^  _ 

whence  the  resultant  of  the  forces  acting  upon  any  one  of  the 
elements  of  a  surface  of  equal  pressure,  is  normal  to  that  surface. 
This  is  the  characteristic  of  what  is  called  a  level  surfacf^  which 
may  be  defined  to  be  any  surface  which  cuts  at  right  angles  the 
direction   of    the  resultant  of  the  forces   which  act  upon  its  particles. 

§2G0. — If  Equation  (420)  be    integrated,  we    have 

\xdx  +  Ydy  -\-  Zdz)  =  C.       •     .     .     .  (422) 


./( 


in  which  C  is  the  constant  of  integration.  The  magnitudes  of  thi.i 
constant  must  result  from  the  dinieu'^ioiis  of  the  surfice,  or  t\\>ui 
the    voluuie    of    the    fluid    it    envehi^;.       Bv    i'-i\irtif    it     dilFereni    and 


2S2  ELEMENTS     OF     ANALYTICAL    MECHANICS. 

snitahle  values,  we  may  start  from  a  single  particle  and  proceed  out- 
wards  to  the  boundary  of  the  fluid,  and  if  the  successive  values 
differ  by  a  small  quantity,  we  shall  have  a  series  of  level  concentric 
strata. 

The  last  possible  value  for  C  will  determine  the  exterior  or  bounding 
surfage  of  the  fluid;  because  this  surface  being  free,  the  pressure  upon  it 
v.ill  be  zero;  the  ditfcreutial  of  the  pressure  from  one  point  to  another 
will,  therefore,  be  zero,  and  the  differential  ccpuition  will  be  that  num- 
bored  (420),  or  that  of  equal  pressure.  livery  free  surface  of  a  fluid  in 
equilibrio  is,  therefore,  a  level  surface. 

§201.— Putting  Equation  (418)    under    the  form 

'^-^  =  Xdx  +  Ydy  +  Zdz, (423) 

we  see  that  whenever  the  second  member  is  an  exact  differential, 
p  must  be  a  function  of  D,  since  the  first  member  must  also  be  an 
exact   differential.     Making,  therefore, 

P  =  F{D), (424) 

in  which  F  denotes  any  function  whatever,  the  above  equation  be- 
comes 

'^-^^^  ^  Xdx  +  Yd^j  +  Zdz;      '     .     .     (425) 

but  for  a  level  surface  or  stratum,  the  second  member  reduces  to 
zero  ;  whence, 

dFi^D)  =  0; 

and    by    integration, 

F{D)  =  C; 

whence,  not  only  will  each  level  stratum  be  subjected  to  an  equal 
pressure  over  its  entire  surflxce,  but  it  will  also  have  the  same 
density  throughout. 

g2G2. — If  the  fluid  be  homogeneous  and  of  the  same  temperature 
throu"-hout,  then  will  D  be  constant,  and  the  condition  of  equilibrium 


i 

MECHANICS     OF     FLUIDS.  2.S3 

simply  requires  that  the  function  Xd.v  +  i^cli/  -)-  Zdz,  Equation 
(419),  shall  be  au  exact  clifFerentiai  of  the  three  independent 
variables  x,  y,  z,  and  when  this  is  not  the  case,  the  equilibrium 
will  be  impossible,  no  matter  Mhat  the  shape  of  the  fluid  mass, 
and    though   it    \\  ere    contained    in    a    closed    vessel. 

But  the  function  above  referred  to  is,  §  133,  always  an  exact 
differential  for  the  forces  of  nature,  which  are  either  attractions  or 
repulsions,  whose  intensities  are  functions  of  the  distances  from  the 
centres  through  which  they  are  exerted.  And  to  insure  the  equi- 
librium, it  will  only  be  necessary  to  give  the  exterior  surface  such 
shape  as  to  cut  perpendicularly  the  resultants  of  the  forces  which  act 
upon  the  surfece  jiarticles.  This  is  illustrated  in  the  simple  example 
of  a  tund>ler  of  water,  or,  on  a  larger  scale,  by  ponds  and  lakes 
which  only  come  to  rest  when  their  upper  surfaces  are  normal  to 
tile  i-esultant  of  the  f  )rce  of  gravity  and  the  centrifugal  force  arising 
li'om    the    earth's   rotation    on    its   axis. 

Ill  the  case  of  a  heterogeneous  fluid  subjected  to  the  action  of  a 
ci'ntral  force,  its  equilibrium  requires  that  it  be  arranged  in  concentric 
level  strata,  each  stratum  having  the  same  density  throughout.  And 
the  e(piilibrium  will  be  stable  when  the  centre  of  gravity  of  the 
wliole  is  the  lowest  possible,  §  138,  and  hence  the  denser  strata  should 
be    the    lowest. 

When  th:^  fluid  is  incompressible,  the  density  may  be  any  function 
whatever  of  the  co-ordinates  of  place.  It  may  be  continuous  or  dis- 
Ciiiilinuous.  When  it  is  given,  the  value  of  the  pi'essure  is  found  from 
Eijuation   (419). 

§  263. — In    compressible    fluids    the    density    and  pressure  are    con- 
nected  by    law,  and    the    former  is  no  longer  arbitrary. 
Dividing  Ec[uation  (418)  by  Equation  (389),  Ave  have 

dp    _  Xdx  +   Ydy  +  Zdz 
~  -  p 

Integrating, 

''Xdx  +   Ydy  +  Zdz 


log_p 


rXdx  +   Ydy  +  Zdz 
J ir^^ +  log  6';  .     .    .  (420) 


2S4;         ELEMENTS     OF     ANALYTICAL    MECHANICS, 
denoting  the  base  oi  the  Naperian  system  by   e,  we  have 


2)  =1  Ce"  f 


(-127; 


and    this  substituted  in  Equation  ('iSG),  gives 

rXil£  +  Ydti  +  Zdz 

J        _ 

^  =  - p ('128) 

These  equations  determine  the  pressure  and  density. 

For  any  surface  of  constant  pressure,  the  exponent  of  e,  in  Equa- 
tion (4:i7),  must  be  constant,  its  dillerentiai  must,  therefore,  be  zero, 
and  all  the  consequences  deduced  from  Eipiatlon  (420)  will  ibllow  ; 
tluit  is,  when  the  fluid  is  at  rest,  it  must  be  arranged  in  level  strata, 
each  stratum  having  the  same  density  throughout,  with  the  addition 
that  the  law  of  the  varying  density  must  be  continuous  by  the  re- 
quirements  of  Mariotte's  law. 

If  the  temperature  vary,  then  will  P  vary,  and  in  order  that 
Equation  (427)  may  be  an  e.\;'iCt  ditfcreiitial,  F  iiiust  be  a  fuiicticui 
oi  xyz,  and  hence.  Equations  (427)  and  (42o),  when  2^  i'^  constant, 
D  will  be  constant;  that  i.>,  each  level  sti-atum  must  be  of  riuitbrm 
temperature  throughout. 

It  is  obvious  that  the  atmosphere  can  never  be  in  equilibrio  ;  fur 
the  sun  heating  unetjually  its  different  portions  as  the  earth  turns 
upon  its  axis,  the  layers  of  equal  pressure,  density  and  temj)t'rature 
c;in  never  coincide.  Hence,  those  perpetual  currents  of  air  known  as 
tlu-  trade  u'iud.s,  and  the  periodical  monsoons ;  also,  the  sea  and  land 
br. 'Oi^es,  variable  winds,  &o.,    &;c. 

■^  '^J  2*)4. — Rest    is  a  relative  term;     when    applied  to  a  particle  of  a 

fluid  mass,  it  means  that  that  particle  preserves  unaltered  its  place  in 
regard  to  the  other  ])articles;  a  condition  consistent  with  a  bodily 
movement  of  the  entire  mass. 

If  a,  liquid  mass  turn  uniformly  about  an  axis,  the  preceding 
equations  will  make  known  its  permanent  figure.  For  this  purpose 
it  will  be  sufficient  to  join  to  the  ft)rces  A",  F,  Z,  the  centrifugal  forca 


MECHANICS     OF     FLUIDS. 


285 


Take    the  axis  z  as  the  axis  of  rotation ;    denote  the  angular  velocity 
bv  9,  and    the  distance  of 
the   particle    M    from    the 
axis  2  by   r\  then  will 

}-2  =  .r2  +  3/- ; 

the  centrifugal  force  of  M 
regarded  as  a  unit  of  mass, 
will  be 

and  its  components  in  the  -^ 

direction  of  x  and  y,  respectively, 


?• .  (p'^ .  —  =  a;  <p  ; 


r  .  (p"^  •  ^-  =  y  9' 


; 


and  these  in    Equation  (418),  give 

dp  =  D.{Xdx  +  Ydy  +  Zdz  -r  ^"-.xdx  ^  9^  ^  .(/?/)••  (429) 

When  the  second  member  is  an  exact  differential,  the  permanent  form 
■will    be    possible. 

I'or    the   free   surface  dp  =  0,  and  we  have 

Xdx  +   Ydy  +  Zdz  +  cpKx.dx  +  92y(Zy  =-.  0-   >  •  {HiO) 
Example  1.— Let    it   be   required     to    find     the    ligure    assumed    by 
the  \ViiQ  surface    of  a    heavy  and    homogeneous  fluid    contained  in    an 
open  vessel  and  rotating  about   a  vertical    axis. 
Here, 

.Y  =  0;      r=0;     Z=  -  g ; 

and    Equation  (430)  becomes 

gdz  =  9^  {x  dx  -f  yd  y). 
Inteai'atincr, 


=  ^_  (.,2  +  _y2)  +  c; 
'2g 


(431) 


^vhich  is  the  equation  of  a  paraboloid  whose  axis  is  that  of  rotation. 


2S() 


ELEMENTS  OF  ANALYTICAL  MECHANICS. 


To  find  the  constant  6',  let  the  vessel  be  a  right  c\lindcr.  with 
circular  base.  Avhose  radius  is  «,  and  denote  by  A  the  height  due  to 
the    velocity   of  the  fluid  at  the  circumference,  then 


a-^  (p- 


\gh, 


and 


h .  r' 


+  0 


(432) 


Denote  by  b  the  height  of  the  liquid  before  the  rotation ;  its 
volume  will  be  Tt  a-  .  b.  Conceive 
the  Avhole  body  of  the  liquid  to 
be  divided  into  concentric  cylin- 
drical layers,  having  for  a  common 
axis  the  axis  of  rotation.  The  base 
of  any  one  of  these  layers  will 
have  for  its  area,  neglecting  c??-^, 
2  7r  r  .  d r,  and  for  its  volume,  taking 
thu  origin  of  co-ordinates  in  the 
bottom  of  the  vessel,  ^ntr.dr.z^ 
which  being  int.grated  between  the 
limits  r  =  0  and  r  =  a,  will  give 
the  whole  volume  of  the  fluid,  and 
hence, 


a^b 


r 


z  r  .  d  r 


replacing  r .  d  r  by  its  value  from  Equation  (4fi2),  and  integrating 
between  the  limits  z  z:^  C  and  z  =.  h  -\-  C,  which  are  the  values 
given    by  Equation  (432)  for  r  =  0    and  r  =  a,  we  find 

C  =b  -  hh, 

and    the   equation  of  the   upper   surflice  becomes 

h. 


The    least   and  greatest   values    for    2r,   are    b  —  \h    and    b  +  ^  h, 
obtained  by  making  r  =  0  and  r  =  a,  so   that  the   depression  rif  tjie 


MECHANICS     OF     FLUIDS. 


287 


liquid  at  the  axis  is  equal  to  its  elevatior.  at  the  surface  of  .he 
cylindrical  vessel,  aud  is  equal  to  half  ihe  height  due  to  the 
velocity  of  the  latter. 

§  '2(So.— Example  2.— Let 
the  fluid  elements  be  attract- 
ed to  the  centre  of  the  mass 
by  a  force  varying  inversely 
as  the  square  of  the  distance. 
Take  the  origin  at  the  cen- 
tre ;  denote  the  distance  to 
the  particle  m  from  that  point 
-by  r,  and  the  intensity  of  the 
attractive  force  at  the  unit's 
distance  by  Jc.     Then  will 

k  a 


_i- 


P  =  m 


r2    ' 


cos  f3.  =  — 


cos  y  =  — 


and 


X=  - 


kx 


,.3    ' 


r  = 


ley 

r 


3    '       '^  —  „3    ' 


which  in  Equation  (430),  give 


k 


—  [x  dx  +  y  d  y  +  z  d  z)  —  cf'  {x  d  x  -\-  y  dy)   —  0, 


k  dr 


^  d  (,r^-  +  y2)  ^  0, 


and  by  integration, 


inakinff 


+  \{^'  +  y')  =  C; 


X"^     +     ?/2     _     j,2   (;og2  ^^ 

in  which  &  denotes   the  angle   made  by  r.  with    the  plane  xy, 


A    J."^^.    ;.2(>os2()    =     a 


288  ELEMENTS     OF     AXALYTICAL    MECHAN-ICS. 

'tin:!     denoting    the    distance    from    the    origin     to    the   point   in    -which 
.1x0  free  surface  cuts  the  axis  z  by  unity,  we  have,  by  making  6  =  90°, 

1    -  ^' 

which    substituted    above,  and    solving    with  respect    to  cos^^,  gives 

1  (p2  .  cos2  (3  =  ^'^''  ~  ^^ (434) 

and   making  r  =  1  +  u,  we  have 

1;  a^  '  cos-  6  = — - 

If  the  angular  velocity  be  small,  then  will  u  be  very  small. 
Developing  the  second  member  "with  this  supposition,  and  limiting 
the    terms    to    the  first    power  of  n.  we  find 

]  (p2  .  cos2  (3  =  /■  {zi  -  3  ^(2). (434)' 

JSeglecting  ou'^,  and  replacing  it  by  its  value,  viz.:  r  —  1,  we 
have    for   a    first   approximation, 

2 


1  +  1^  '  cos2  6. 
'2  k 


From  Equation  (434)',  we  find 


©-•cos^a         ^ 
u  — — h  o  u^, 

and   this   in   the    equation 

r  —  \  ^  V, 
gives 

r   -\    +    ~y  C0S2  &   +   3  m2  ; 

Zk 

(p*  •  cos**  () 
and  replacing  u^  by  its  approximate  value  — — — — -.    above,  by  neg- 
lecting 3  U"^  we   have 

(p2  ^   ,  3  p«  •  C0S4  /» 

for   the   polar  equation  of  the   meridian  section. 


MECHANICS     OF    FLUIDS  289 

Comparing    this    Avitli   the    equation 

1  +  i-e-  C032  ^  +  I .  e*  .  cos*  6  +  &c., 


they   become  identical    by    neglecting    the  higher  powers    and  making 


The  free  surface  of  the  fluid  approximates  therefore  very  closely 
to  an  ellipsoid  of  revolution  of  which  the  eccentricity  of  its  meridian 
section  is  equal  to  the  square  root  of  the  quotient  arising  from 
dividing  the  centrifugal  force  at  the  unit's  distance  from  the  axis 
of  rotation,  by  the  force  of  attraction  at  an  equal  distance  from  the 
centre. 

PKESSUKE    OF   HEAVY   FLUrDS. 

§266. — When  a   fluid   contained   in    any    vessel  is    acted   upon   by 
its  own  weight,  if    the   axis  z  be    taken   vertical 
and  positive  downwards,   then  will 

X=0;      Y=0;     Z^ff; 

and    Equation  (418)  becomes,  after  integrating, 

p  =Dgz  +  C; 

and    assuming    the   plane   xy   to    coincide    with 

the  upper    surface    of  the   fluid,  which   must,    when    in    cquilibrio,   be 

horizontal,  we   have,  by  making  2  =  0, 

P'  -  C; 

in  Avhich  2^  denotes   the   pressure    exerted  upon   the   unit  of  the  free 
surface.     Whence, 

p  —  p'  =  D  .g  .z. (435) 

The   first  member    is    the  pressure    exerted    upon  a    unit    of  surflice, 
every  point   of  which  unit  having  a   pressure  equal   to  that  sustained 

In-    the    element   whose    co-ordinate   is    z. 

19 


290 


ELEMENTS     OF     ANALYTICAL    MECHANICS 


If  p'  =:  0,   tlicn    will 

P^Dg^^; (436) 

and  denoting  by  h  the  ai-ea  of  the  surface  pressed,  and  by  dh^  the 
element  of  this  surface,  whose  co-ordinate  is  2,  we  have,  Equation 
(397),  for  the   pressure    upon    this    element   denoted    by   jy^, 

P.  =  Dg.z.db, 

and  the  same  for  any  other  element  of  the  surface ;  whence,  deno- 
ting   the    entire   pressure  by  P,    we    shall    have 

P  ^-Lp^  :zz  Dg.I.z.dh.     .     .     .'.     .     .(437) 

But  if  z^  denote  the  co-ordinate  of  the  centre  of  gravity  of  the 
entire  surface  Z>,  then    will,  Equations  (91), 

I,z  .d  b  =z  bz^, 
and 

P  =  Df/.b.z^. (438) 

Now  b  z,  is  the  volume  of  a  right  cylinder  or  prism,  whose  base 
is  6,  and  altitude  s^ ;  Dy.b.z^  is  the  weight  of  this  volume  of 
the  pressing  fluid.  Whence  we  conclude,  that  the  inesKure  exerted 
upon  avy  siirface  by  a  heavy  fluid  is  equal  to  t//e  weight  of  a  cyliii- 
drical  or  jjrismutic  column  of  the  fluid  loliose  base  is  equal  to  the 
surface  pressed,  and  whose  altitude  is  equal  to  the  distance  of  the  cen- 
tre   of  gravity    of  the    surface  bcloiu  tlie    tqrper   surface    of  the  fluid. 

When  the  surface  pressed  is  horizontal,  its  centre  of  gravity  will 
be  at  a  distance  from  the  upper  surface  equal  to  the  depth  of  the 
fluid. 

This  result  is  wholly  independent  of  the  quantity  of  the  pressing 
fluid,  and  depends  solely  upon  the  density  of  the  fluid,  its  height,  and 
the    extent    of  the    surface    pressed. 

p^xamjyle  1.  —  Required  the  pressure 
against  the  inner  surface  of  a  cubical  ves- 
sel  fdled  with  water,  one  of  its  faces  being 
horizoiUal.  Call  the  edge  of  the  cubo  a. 
the  area  of  each  flxce  will  be  «^  the  dis- 
tance of  the  centre  of  gravity  of  each 
vertical     face    below   the    upper   surfice  will    be  ^a.    and    that   of  tin 


lower    face   a ; 
gives, 

Again, 


MECHANICS    OF    FLUIDS.  291 

whence,    the    principle     of    the     centre    of    gravity 


and   these,  substituted  in  Equation   (438).  give 

P  =-  B  .g-b.z^  ^  D.(/  .Za\ 

Now  D  g  X  13  =  Z>(/,  is  the  weight  of  a  cubic  foot  of  water  =z  62,5 
lbs.,  whence, 

lbs. 

P  =  62,5  X  3a3. 
Make   a  —  1  feet,  then  will 

lbs. 

P  =  62,5  X  3  X  (7)3  =  64312,5. 

The  weight  of  the  water  in  the  vessel  is  62,5  a^,  yet  the  pressure 
is  62,5  X  3a-^,  whence  we  see  that  the  outward  pressure  to  break 
the  vessel,  is   tliree  times   the    weight   of  the    fluid. 

Example  2. — Let  the  vessel  be  a  sphere  filled  with  mercury,  and 
let  its  radius  be  R.  Its  centre  of  gravity  is 
at  the  centre,  and  therefore  below  the  upper 
surface  at  the  distance  R.  The  surface  of  the 
sphere  being  equal  to  that  of  four  of  its 
great  circles,  we  have 

b  =  4*^22. 


whence. 


and.  Equation  (438), 


b.z,  =  4*i23; 


P  =  ^nr.D.g.R^ 


The  quantity  Dg  x  V  —  Pg,  is  the  weight  of  a  cubic  foot  of 
mercury  =  843,75  lbs.,  and  therefore,  substituting  the  \alue  of 
ff  =  3,1416, 


4  X  3,1416  X  843,75  .  R\ 


292 


ELEMENTS  OF  ANALYTICAL  MECHANICS. 


Now   suppose    the   radius    of    the    sphere    to    be    two    feet,    then   wilJ 
B^  =z  8,  and 


P  =  4  X  3.1416  X  843,75  X  8 


lbs. 

84822,4. 


The  volume  of  the  sphere  is  f  ■?  IP ;  and  the  weight  of  the  con- 
tained mercury-  will  therefore  be  ^-TrE^gD—W.  Dividing  the 
whole   pressure   by  this,  we  find 

F 


W 


=1  3 


whence   the    outward    pressure  is  three   times  the  weight  of  the  fluid. 

Example  3, — Let    the  vessel   be   a    cylinder,  of    which    the    raOius 
r  of  the  base   is   2,    and    altitude    /,   6    feet.     Then    will 

b.z,  =  'xrl{r  +  I)  ^  3,1410  X  2  X  o  X  8; 

which,  substituted  in  Equation  (438), 

P  =  301,5936  X  Dg, 


and 


whence, 


W 


3,1410  X  2-  X  0  X  X»:/  =  75,398  X  D(j\ 
301,5930  X  Dq 


P_ 

W 


75,3984  .  JJg 


that    is,  the  pressure    against  this    particular  A^essel    is  four   times   the 
A'eight   of  the    fluid. 

§207. — The  point  through  which  tlie  resultant  of  the  pressure 
upon  all  the  elements  of  the  surface 
passes,  is  called  the  centre  of  j^ressure. 
Let  EIF  be  any  phuie,  and  MN 
the  intersection  of  this  plane  produced 
with  the  upper  surface  of  the  fluid 
which  presses  Mgainst  it.  Denote  the 
area  of  any  eleinentary  portion  n  of 
the  ])hme  EIF  by  dh  \  and  let  m  be 
the  j)n)jectioii  of  its  place  upon  the 
upper  surface  of  the  fluid;  draw  mM 
pei*pendicular  to  J/iV,  and  join  n  with  i/by    the  right  line  n  3/,  thw 


/b  - 


MECHANICS     OF    FITTfDS.  203 

latter    vill    also  be  pi;r]»eii(liciil;ir   to  M N,  and    the  angle  n  Mm  will 
measure    the    inclination    of    tlie    plane    £  IF  t<>    the    surface    of  the 
fluid.      Denote  this  angle  by  9,  the  distance  m  n  l>y  h\  and  J/n  by  r' 
then    will 

h'  :=  r'  sin  9  ; 
the   pressure   upon    the    element   d  b, 

D  rj  .r'  sin(p  dh\ 

its    moment   with    reference    to   the   line  MN^  ^y 

D g r''^  am  (p  .  dh\  ^ 

and  for    the    entire  surface,  the    moment   becomes 
D  (J  .  sin  9  .  2  r'"^  db. 

Denote  by  r  the  distance  of  the  centre  of  gravity  of  the  surface 
pressed  from  the  line  M  N,  its  distance  l)elow'  the  upper  surface  of 
the  fluid  wdll  be  r .  sin  9 ;  and  the  pressure  upon  this  surface  will  be 

jf)  y  .  ?•  sin  9  .  i  ; 

and  if  I  denote  the  distance  of  the  centre  of  pressure  from  the 
line  M N,  then  will 

I)  g  .  r  sin  9  .b.l  r=i  D  g  •  sin  9  .  2  r'2 .  rf  J, 

from  which  w'e  have,  ^    -    ^  ,     ^>    ^     —      ,     

..li:!#; T.     .     (430) 


whence.  Equation  (238),  the  centre  of  pressure  is  found  at  the  centre 
of  percussion  of  the   surface  pressed.   ''<^  '■'  ''*■    '^^Z^-**-*^'-  "^ 


§  268. — The  principles  which  have  just  been  explained,  are  of 
great  practical  importance.  It  is  often  necessary  to  know  the  pre- 
cise amount  of  pressure  exerted  by  fluids  against  the  sides  of  ves- 
sels and  obstacles  exposed  to  their  action,  to  enable  us  so  to  adjust 
the  dimensions  of  the  latter  as  to  give  them  suflicieiit  strength  to 
resist.  Reservoirs  in  which  considerable  quantities  of  Avater  are  col- 
lected and  retained  till  needed  for  purposes  of  irrigation,  the  supply 
of  citie?    and    towns,  or   to    drive  machinery  ;    dykes  to  keep   the  sea 


2[)i 


ELEMENTS  OF  ANALYTICAL  MECHANICS. 


and  kikes  from  inundating  low  districts ;  artificial  embankments  con- 
structed along  the  shores  of  rivers  to  protect  the  adjacent  country 
in  times  of  freshets ;  boilers  in  which  elastic  vapors  are  pent  up  in 
a  high  state  of  tension  to  propel  boats  and  cars,  and  to  give  motion 
to    machinery,  are  examples, 

g269. — As  a  single  instance,  let  it  be  required  to  find  the  thick- 
ness of  a  pipe  of  any  material  necessary  to  resist  a  given  pres- 
sure. 

Lot  ABC  be  a  section  of  pipe  perpen- 
dicular to  the  axis,  the  inner  surflice  of 
wliich  is  subjected  to  a  pressure  of  2^  i^ounds 
on  each  superficial  unit.  Denote  by  R  the 
radius  of  the  interior  circle,  and  by  I  the 
length  of  the  pipe  parallel  to  the  axis ; 
then  will  the  surface  pressed  be  measured 
b}-  2  <  i?  .  / ;  and  the  whole  pressure  by 
^'zR.1.2). 

My  virtue  of  the  pressure,  the  pipe  will ,  stretch ;  its  radius  will 
become  R  -f  t/i?,  the  path  described  by  the  pressure  will  be  d  R^ 
and    its    quantity  of  work 

2n(  R.l.2>d  R. 

The  interior  circumference  before  the  pressure  was  2irR,  afi;erwards 
2'r{R  +  dR),  and  the  path  described  by  resistance,  2'K'dR.  And 
if  R  denote  the  resistance  which  the  material  of  the  pipe  is  capable 
of  opposing,  to  a  stretching  force,  without  losing  its  elasticity  over 
each  unit  of  section,  ^  the  thickness  of  the  pipe,  then,  by  the  prin- 
ciple of  the    transmission  of  work,  must 


IV  hence, 


2^^  .  B.l.dR.t  ^2'n  R.l.2).dR', 
Rp 


t  — 


B 


The    value  of  p    is    estimated    in    the    cjise    of  water    jiressure   by 
the   rules  just  given.     That  in   the  case  of  steam  or  condensed  gases, 


MECHANICS     OF    FLUIDS. 


295 


by  rules  to  be  given  presently.  The  value  of  B  is  readily  obtained 
from  Taole  I,  giving  the  results  of  experiments  on  the  strength  of 
materials. 


^ 


EQUILIBKIUM   AIJD    STABILITY   OF   FLOATING  BODIES. 


§  270. — When  a  body  is  immersed  in  a  fluid  it  is  not  only 
acted  upon  by  its  own  weight,  but  also  by  the  pressure  arising  from 
the  weight  of  the  fluid,  and  the  cii'cumstances  of  its  rest  or  motion 
will   be    made    known    by   Equations  (yl)  and  {B). 

Let  ED  be  the  body  ;  take  the  plane  x  y  in  the  plane  of  the  up- 
per surface  of  the  fluid, 
supposed  at  rest,  and 
the  axis  of  z  therefore 
vertical.  Denote  by 
h  the  entire  surface 
of  the  body,  and  by 
d  h.  one  of  its  elements, 
whose  co-ordinates  of 
position  are  x  y  z.  The 
jiri'ssure  upon  this  ele- 
ment will  be 


D.g 


d  h. 


in  which  D  is  the  density  of  the  fluid,  and  rj  the  force  of  gravity. 

This  pressure  is,  g  248,  normal  to  the  surface,  and  denoting  by 
a,  /3  and  y,  the  angles  which  this  normal  makes  with  the  axes  xyz. 
respectively,  the  components  of  the  pressure  in  the  direction  of  these 
axes  will  be 

D  '  g  .z  ,db  .  cos  a  ;     D  .  g  .z  .dh .  cos  /3  ;     D  .g  .z  .dh  .  cos  y. 

Similar  expressions  being  found  for  the  components  of  the  pressure  on 
other  elements,  we  have,  by  taking  their  sum, 

D  g  .'S.Z  .db  .  cos  a  ;     D  g  .1.  z  .dh  .  cos  /3  ;     Dg  .1.  z  .  db  .  cos  y. 

But   ao.coscc,    c/o.coSjS,    and    db.cosy.    are   the  projections    of  the 
area  db  on  ta  ■  co-ordinate  planes  z  y,  zx  and  .r  y,  respectively,    and 


296  ELEMENTS     OF    ANALYTICAL    MECHANICS. 

"E  z  .  dh  .  cos  a|2  z  .  d  b  .  cos  (3,1,  z  .d  b  .  cos  ^J  Qii'sJ-  volume^s)  of'^  right 

prisml)  ^vli()sc     bases  (are"]  projection^ i  of    the    entire    surfuce     pressed 

upon    the /same  co-orcliiiate   plant^j  and   of   which   ihe   altitudej  of  each  ( 

is   the   depth   of  the   coinuion   centre  of  gravity  of  the   elements  ot^its 

base  stt^Hft»ei^ge^''iTriiic---tirpl4TyT>fTl)eir''co^^  miifacu  t'li^MwDl^s. 

Whence    we   conclude,   that    ihe  comj)oite/tt    of    the  ^^rcs.y^j-e    on  awQ 

w^'    surface,  estimated  in  any  direction,  is  equal    to   the  pressure  on  so  muck 

■    1/     of  that   surface  as  is  equal    to  its  2»'oJection  on  a  'plane  at  right  angles 

'     I     to    the   given    direction. 

The  cylinder  or  prism  which  projects  an  element  on  one  side  of 
the  body  will  also  project  an  element  situated  on  the  opposite  side ; 
these  jjrojections  will,  therefore,  be  equal  in  extent,  but  will  have 
contrary  signs,  for  the  normal  to  the  one  will  make  an  acute,  and 
to  the  other  an  obtuse  angle  with  the  axis  of  the  plane  of  jirojection. 
When  these  projections  are  made  upon  any  vertical  plane,  the  value 
of  z  will  be  the  same  in  both,  and  hence,  for  each  positive  product, 
z  .  db  .  cos  u.  and  z  .  d  b  .  cos  /3,  there  will  be  an  equal  negative  product ; 
therefore, 

I)  g  .  I.  z  .db.  cos  a  z=z  2  P  cos  a  =  0;  Dg  .'Ez.db  .  cos /3  =  2  Pcos /3  =  0. 

That  is,  tlie  sum  of  the  horizontal  pressures  in  the  directions  of 
X  and  y,  and  therefore  in  all  horizontal  directions,  will  be  zero  ;  and 
the   first   and    second   of  Equations  (1:20),  give 

2  m  •  -—  1^  0  :    2  vi  ■  —4  =  0  ; 
dt"  '  dt-  ' 

or,  which  is  the  same  thing,  there  can  be  no  horizontal  motion  ol 
translation    from    the    fluid    pressure. 

When  the  projections  of  opposite  elements  are  made  upon  a 
horizontal  plane,  they  will  still  be  equal  Avith  contrary  signs,  the 
normal  t(J  the  elements  on  the  lower  side  making  obtuse,  wliile  the 
normals  to  the  elements  above  make  acute  angles  with  the  axis  z; 
but  the  cori-esponding  values  of  z  will  difier,  and  by  a  length  equal 
to  that  of  the  vertical  filament  of  the  body  of  which  these  elements 
form    the    opposite    bases,  and    hence 

Dg.-Ez.db.eo^y^  D  g  .S  {z'—z^)  d  b  aosy  zzz  —  D  gl.cd  bcosy  ■  {-liO) 


MECHAXICS    OF    FLUIDS.  297 

in  which  z'  denotes  the  oi'dinate  for  the  uj^per,  and  z,  that  for  the 
lower  element  in  the  same  vertical  line,  and  c  the  distance  between 
the  elements;  and   the   third  of  Equations  (120)  becomes 

2  \P  cos  y  —  m-  -j^J  =  -%  —  D  ff -^  c  ■  d  b  •  cosy  —  "Lm-  — -^  =  0. 

But  Ic.dh.co^y  is  the  volume  of  the  immersed  body  which  is 
obviously  equal  to  that  of  the  displaced  fluid;  also  D  (j  .1  c  .d  b  .Qosy 
is  the  weight  of  the  displaced  fluid;  and  J/y  that  of  the  bo. I  v. 
Denoting  the  volume  of  the  body  by  V,  its  density  by  D\  ;he 
above    may  be  written 


Now,  Avhen 


V'D'g  -   V  Dg  -Zm^'^  =  0.      •     •     •     (441) 


V  D'  g  -   V'Dg  =  0, 
J)  =  D\ 


then  will 


d-z 

and    there   can    be   no    vertical   motion    of  translation    fi -m    the    fluid 
pressure  and   the    body's    weight. 
When  D'  >  D,  then  will 

2m.^=(Z)'-i>)r'.^; 

and    the    body  will    sink  with    an    accelerated    motion. 
When  JJ'  <  D,  then  will 

2«.^=  -{D'-D)   V'.g, 

and    the    body  will  rise  vrith  an  accelerated   motion    till 

d^z 
^m-  -j-^  =  V  L'  y  -    V  Dg  =r  0  ;      •     •  C^!4->) 


29S 


ELEMENTS  OF  ANALYTICAL  MECHANICS. 


in  which  V  denotes  the  volume  A  B  C,  of  the 
fluid  displaced.     At    this    instrait  we   have 

V  D'cf  =  V  Lfi;  '     '     •     (443) 

and  if  the  body  be  brought  to  rest,  it  will 
reiiiain  so.  That  is,  the  body  will  float  at  the 
surface  when  the  weight  of  the  fluid  it  dis- 
places is  equal  to  its  own   weight. 

The  action  of  a  heavy  fluid  to  su])port  a  body  wholly  or  partly 
immersed  in  it,  is  called  the  huoijaut  effort.  The  intensity  of  the 
buoyant  effort  is  equal    to  tJte  iveight  of  the  fnikl  dis-placed. 

Substituting  the  values  of  the  horizontal  and  vertical  components 
of  the    pressures  in  Equations  (US),  and   reducing   by   the  relations, 


D  ij  .'S-  c  .  d  h  .  cos  y  ^.x'  =  D  g  .  V .  x  ; 
Dg  .Zc  .d  b  .  cos  7  .  ?/'  =  Z)  ff  .  V  .y] 


(444) 


in  which  x  and  y  are  the  co-ordinates  of  the  centre  of  gravity  of  the 
displaced  fluid  referred  to  the  centre  of  gravity  of  the  body,  we  find 


2  m 


x'.d'^y' 


d^-x' 


df^ 


=:   0 


z'-d-^x' 
I,  m-  - 


x'  •  d^  z' 


dt- 


=  Dy.V-x;    } 


2  m  .  -L- -j^ -'^  ^  -  Dy.\  .y. 


(445) 


Equations  (444)  show  that  the  line  of  direction  of  the  buoyant 
efl'ort  passes  through  the  centre  of  gravity  of  the  displaced  fluid. 
This  jjoint  is  called  the  centre  of  buoyancy.  And  from  Equations 
(445),  we  see  that  as  long  as  x  and  y  are  not  zero,  there  will  be 
an  angular  acceleration  about  the  centre  of  gravity.  At  the  instant 
X  —  0  and  y  —  0,  that  is  to  say,  wlien  the  centres  of  gravity  of 
the  body  and  disphiced  fluid  are  on  the  same  vertical  line,  this 
acceleration  will  cease,  and  if  the  body  were  brought  to  rest,  it 
would   ha\o   no    tendency   to    rotate. 

To    recajiitulate,  we  fuid, 


MECHANICS     OF     FLUIDS. 


299 


1st.  That  the  pressures  upon  the  surface  of  a  body  immersed  in 
a  hcaty  fuid  have  a  sint/le  resultant,  called  the  buoyant  effort  of  the 
fuid,  and   that    this  resultant    is   directed    vertically    upwards. 

•2ci.  That  the  buoyant  effort  is  equal  in  intensity  to  the  weiyht  of 
the  fiuid   displaced. 

3(3.  That  the  line  of  direction  of  the  buoyant  effort  2)<:isses  throuyh 
the  centre    of  gravity  of  the  displaced  fluid. 

4th.    That  the    horizontal  pressures    destroy    one    another. 

§271. — Having  discussed  the  equilibrium,  consider  next  the  sta 
bility  of  a  floating  Itody.  The  density  of  the  Ijody  may  be  homo- 
geneous or  heterogeneous. 
Let  A  B  CD  be  a  section 
of  the  body  by  the  upper 
surface  of  the  fluid  when 
th'.-  hddy  is  at  rest,  G 
its  centre  of  gravity,  and 
11  that  of  the  fluid  dis- 
phiced.  Denote  by  V  the 
vnkiine  of  the  displaced 
Ouid,  and  by  M  the  mass 
of   the    entire   body.     The 

body  being   in  equilibrio,  the  line   6^JTwill  be  vertical,  and  denothig 
the    density  of  the    fluid    by  D,  we    shall   have 


M  =  D.  V. 


(440) 


Stippose  the  section  ABCD  either  raised  above  or  depressed 
below  the  surface  of  the  fluid,  and  at  the  same  time  slightly  careened  ; 
also  suppose,  when  the  body  is  abandoned,  that  the  elements  have 
a  slight  velocity  denoted  by  u,  u\  &c.  Now  the  question  of  sta- 
i)i]ity  will  consist  in  ascertaining  whether  the  body  will  return  to  its 
former  position,  or  will    depart  more    and  more  from    it. 

The  free  surface  of  the  fluid  is  called  the  ^j>/a?ie  of  floatation, 
and  during  the  motion  of  the  body  this  plane  will  cut  from  it  a 
variable   section. 

Let   A'  B'  C  I)'  be  one  of  these  sections   at  any  given  instant  cf 


300  ELEMENTS     OF     ANALYTICAL     MECHANICS. 

time  ;  A  B"  CD",  auotlicr  variaLIc  section  of  the  body  by  a  hori- 
zontal  plane  thi'ough  the  centre  of  gi'avity  of  the  primitive  seetior. 
A  U  CD,  and  A  C  the  intersection  of  the  two.  Denote  by  (?  the 
inclination  of  these  two  sections,  and  by  ^  the  vertical  distance  of 
AB"CD'\  from  the  plane  of  floatation,  Avhich  now  coincides  with 
A'  B'  CD',  this  distance  being  regarded  as  negative  or  positive,  ac- 
cording as  A  B"  C  D"  is  below  or  above  the  plane  of  floalaticm. 
The  variable  quantities  t)  and  ^  will  be  suj)posed  very  small  at  the 
instant  the  body  is  abandoned.  Will  they  continue  so  dui'ing  the 
whole  time  of  motion  1 

From  the  ))rinciples  of  living  force  and  quantity  of  work,  we  have, 
Equation    (I"21), 

fuKdAf  =  2f{Xdx  +   Vdy  +  Zds)  +   C. 

The  forces  acting  are  the  weights  of  the  elements  dM  and  the  verti- 
cal pressures,  the  horizontal  pressnres  destroying  one  another ;  whence, 
X  =  0,     F  =  0,    and 

y«2  d  M  =  2  fzdz  -\-C=2^Zz+C.     •     .      (447) 

The  force  wdiich  acts  upon  an  element  above  the  plane  of  floata- 
tion is  its  own  weight,  and  the  force  which  acts  upon  any  element 
below  that  plane  is  the  difference  between  its  own  weight  and  that 
of  the  fluid  it  displaces ;  the  first  will  be  (/  .  d  i/,  and  the  second, 
g.D  .d  V,  in  which  d  V  is  the  volume  o(  d  M;    whence, 

:^Zz  z^fg.z.dM-fyD.z.dV.      -     •     •     (448) 

But.  (h-awing  from  the  centre  of  gravity  G,  of  the-  body,  the  perpen- 
dii-iiiar  (r  I'J,  to  the  plane  of  floatation  A'  B'  C  D\  and  denoting  GE 
bs    i'^,   v,e  have 

Jg.s.dM^gMz^. 

The  integral  A  7)  .£-.(/  r,  will  be  divided  into  two  parts,  viz:  ona 
relating  to  the  volume  of  the  body  btdow  A  B  CD,  or  the  V(dume 
ijiunersed  in  a  state  of  rest,  and  the  other    that     comprised    between 


MECHANICS    OF     FLUIDS.  301 

AD  CD  and  the  plane  of  floatation  A' B' C  D\  Avhen  the  body  is  in 
motion.  Di'iiote  by  y  D  V z\  the  vahic  of  the  first,  in  which  z' 
denotes  the  variable  distance  ///',  of  the  centre  of  gravity  //,  of 
the  volume  F,  from  the  phuie  of  floatation  A'  B'C'D'.  And  repre- 
senting for  the  instant  by  h  t.ie  value  of  the  integral  I  zdV,  com- 
prehended between  the  planes  .4  i?  CZ>  and  A' D'  C  D',  g  D  h  \\\\\ 
be   the    second    part;  and  Equation  (447)  becomes 


f 


ir-dM  =:  2r/.3fz^  —2cjDVz'-  2gDh  +  C.  ■  -  (449) 

The  line  G  S,  being  perpendicular  to  the  plane  A  B  C  J),  the  angle 
which  it  makes  with  the  line  G  E  is  equal  to  &.  and  denoting  the  dis- 
tance  G  H  by  a,  we  have 

z^  —  z'  ±  a  cos  ^  ; 

the  upper  sign  being  taken  when  the  point  G  is  below  the  point 
//,  and  the  lower  when  it  is  above.  This  value  reduces  Equation 
(440)    to 


/' 


??"  dM  =  ±:1g  D  Va  cos  &  —  2  g  D  k  +  C.    •    •    •  (4.'')0) 

Let  us  now  find  the  integral  //.  For  this  purpose,  conceive  the 
area  A  BCD  to  be  divided  into  indefinitely  small  elements  dmoted 
by  dX,  and  let  these  be  projected  upon  the  plane  of  floataiiou, 
A'  B'  C  D'.  The  projecting  surfaces  will  divide  the  volume  com- 
prised between  these  two  sections  into  an  indefinite  number  of 
vertical  elementary  prisms,  and  these  being  cut  by  a  series  of  hori- 
zoiital  planes  indefinitely  near  each  other,  Avill  give  a  series  of  ele- 
mentary volumes,  each  of  which  will  be  denoted  by  d  V,  aial  we 
shall    have 

d  V  —  dz  .  d\.  cos^  ; 

whence,  for   a    single  elementary   vertical    prism, 

fz  dV  -  fzdz.dX.  cos  d  =  A  (2)2  .cosd.dX; 
hi  which  (z)  denotes  the  mean  altitude  of  the  prism,  and  consequently 

h  —  I  cos  (3  .  J'izy  .  d  X, 
which    must    be    extended    to    embrace    the    entire    surface  A  B  CD. 


302  ELEMENTS-   OF     ANALYTICAL     MECHANICS. 

The  value  of  (2:^  is  composed  (,  f  two  parts,  viz.:  one  comprised 
between  the  parallel  sections  A' B'  C  D'  and  AB"CD",  and  which 
has  been  denoted  by  ^;  the  other  comprised  between  the  base  d'K 
and  the  second  of  these  planes,  and  which  is  ec|^ual  to  / .  sin  ^,  de- 
noting by  I  the    distance  of  d'K  from    the  intersection  A  C ;    whence, 

(z)  =  ^  +  Z.sin^, 

in  Avhich  I  will  be  positive  or  negative  according  as  d\  happens  to 
be  below  or  above  the  plane  AB"  C D".  Substituting  this  in  the 
value  of  A,  and  recollecting  that  ^  and  d  are  constant  in  the  inte- 
gration, we   find 

h  —  ^^".cos6  .J  dX  +  ^sin&cosd  J  IdX  +  ^sm^S  .  cos6  fp  d\. 

Denote  by  b  the  area  of  A  B  C D,  or  the  value  of  /  dX.  The 
line  A  C  passing  through  the  centre  of  gravity  of  ABCD,  we  have 
I  IdX  =  0.  And  denoting  by  k\  the  principal  radius  of  gyration 
of  the    surface  b,  in    reference    to    the    axis  A  C, 

fpdx  =  bk-;^. 


in    which  the   value  of  k^    is    dependent    upon    the  figure    and    extent 

of  the  surface  ABCD,  and  upon  the  position  of  the  line  AC. 
Whence, 

h  =  lb.  cos  0  (^2  +  j^.2  sin2  S).  ....     (4r>l  ) 

Taking 

sin  &  =  &  —  ^^—5  +  &c ;     cos  5  =  1  —  — -^  +  &c. 

Neglecting  all  the  terms  of  the  third  and  higher  orders,  substitut- 
ing in  the  value  of  A,  and  then  in  Equation  (450)  we  find,  after  trans- 
posing and    includuig   the   term    ±2ffl)  Vd,  in  the  constant  C, 

fu^.dM+  rjnh^^  +  (6/-,2  ±  Ya^  ()2l^  c'.  •  •  .(452) 

Now  the  value  of  the  constant  C  depends  upon  the  initial  values 
of  u,  6  and  ^ ;  but  these  by  hypothesis  are  very  small ;  hence  C, 
must  also   be  very   small.     As  long  as  the  second  term  of  the  first 


MECHANICS     OF     FLUIDS.  303 

member  is  positive,  /  ifi  d  M  iinist  remain  very  small,  since  it  is  essen- 
tially positive  itself,  and  being  increased  by  a  positive  quantity, 
the  sum  is  very  small.  Hence  ^  and  &  must  remain  very  small. 
But  when  the  second  term  is  negative,  which  can  only  be  when 
hlc^"^  ±  V a.,  is  negative  and  greater  than  i  ^,  the  value  o?  I  ii^  d M 
may  increase  indefinitely ;  for,  being  diminished  by  a  quantity  that 
increases  as  fast  as  itself,  the  difference  may  be  constant  and  very 
small.  Hence,  ^  aiid  6  may  increase  more  and  more  after  the 
body    is    aliandoned   to   itself,    and   finally    it    may   overturn. 

The  stability  of  the  equilibrium  depends,  therefore,  upon  the  sign 
of  b  k\"  ±  Va;  the  equilibrium  is  always  stable  when  this  quantity  is 
positive ;    it    is  unstable    when    it   is    negative   and    greater    than  b  =_. 

r  ^" 

The  value  of  b  k^"  ^=  i  PdX,  must  always  be  positive,  since  all  its 
elements  are  positive ;  the  value  of  ±  Ta  becomes  negative  when 
the  centre  of  gravity  of  the  body  is  above  that  of  the  displaced 
fluid,  in    which    case    the  stability    requires   that 

bk;^>  Va,     or,  V  >-^- 

When  the  centre  of  gravit}'  of  the  body  is  below  that  of  the  dis- 
pkced    fluid,  the   sign    of    Va   is   positive. 

Whence  we  conclude  that  the  equilibrium  of  a  body  floating  at 
the  surface  of  a  heavy  fluid,  will  be  stable  as  long  as  the  centre 
of  gravity  of  the  body  is  bfhjw  that  of  the  displaced  fluid;  that 
it  will  also  be  stable  about  all  lines  A  C,  Avith  reference  to  which 
the  principal  radius  of  gyi-ation  of  the  section  of  the  body  by  the 
plane  of  floatation  squared,  is  greater  than  the  volume  of  the  dis- 
placed fluid  multiplied  by  the  distance  between  the  centres  of 
gravity  of  the  displaced  fluid  and  that  of  the  body,  when  the  latter 
i;  in  equilibrlo,  divided  by  the  area  of  the  section  of  the  body 
by  the  plane  of  floatation.  When  this  condition  is  not  fulfilled,  the 
equilibrium  may  be  unstable.  A  ship  w'hose  centre  of  gravity  is 
above  that  of  the  water  she  displaces,  may  overturn  about  her  longer, 
but  not  about  her  shorter  axis. 

g2T2.— A   line  x?7v' through   i!ie  centre  of  gravity    G  of  the  body 


30-i  ELEMENTS     OF     ANALYTICAL    MECHANICS. 

iind  which  is  vertical  when  the  body  is  in  equilibrio,  is  called  a  line 
of  rest.  A  vertical  line  H' M 
through  the  centre  of  gravity 
//'  of  the  displaced  fluid,  is 
called  a  line  of  siijq^ort.  The 
point  i/,  in  M'hieh  the  line  of 
support  cuts  the  line  of  rest, 
is  called  the  metacentrc.  The 
body  will  be  in  equilibrio 
when  the  line  of  rest  and  of 
support  coincide.  The  equi- 
librium will  be  stable  if  the  metacentrc  fall  above  the  centre  of 
gravity  ;    it  may  be  unstable  if  below, 

§273. — When  the  cquilil)rium  is  stable,  and  the  body  is  disturbed 
and  then  abandoned  to  the  action  .jf  its  own  weight  and  that  of 
the  fluid  pressure,  it  will,  in  its  ellbrts  to  regain  its  place  of  rest, 
oscillate   about   this  position,  and  finally  come    to    rest. 

The    circumstances  of  those    oscillations  about    the  centre  of  gravity 
of  the   body  will    readily  result  from  Equations  (445). 


-I 


SPECIFIC     GKAVITY. 


§  274. — The  specific  r/raviit/  of  a  body,  is  the  weight  of  so  much 
of  the    body,  as  would    be    contained  under    a    unit    of  volume. 

It  is  measured  by  the  quotient  arising  from  dividing  the  weight 
of  the  body  by  the  weight  of  an  equal  volume  of  some  other  sub- 
stance, assumed  as  a  standard  ;  for  the  ratio  of  the  weights  of  equal 
volumes  of  two  bodies  'neing  always  the  same,  if  the  unit  of  volume 
of  each  be  taken,  and  one  of  the  bodies  become  the  standard,  its 
weight   will    become    the    unit  of  weight. 

Tlie  term  deiisitij  denotes  the  degree  of  proximity  among  the 
pai'ticles  of  a  body.  Thus,  of  two  bodies,  that  will  have  the  greater 
density  which  contains,  under  an  equal  volume,  the  greater  nuinber 
of  particles.  The  force  of'  gravity  acts,  within  moderate  limits, 
equally    upon    all    elements    of    matter.      The    weight   of  a    substance 


MECHANICS     OF     FLUIDS.  305 

is,  therefore,  directly  })roportional  to  its  density,  and  the  ratio  of 
the  -weights  of  equal  volumes  of  two  bodies  is  equal  to  the  ratio 
of  their  densities.  Denote  the  weight  of  the  first  by  W,  its  density 
by  D,  its  volume  by    V,  and   the  force    of  gravity   by  ff,  then  will 

and  denoting  the  like  elements  of  the  other  body  by  W] ,  D^  and 
V^ ,  we   have 

Dividing   the   first   by  the  second. 


w 

(jDV          DV 

w, 

"  9l^,V.  ~  ,^^V, 

and   making  the  volumes 

equal. 

W        D 

W,        D. 

(453) 


Now  suppose  the  body  Avhose  weight  is  W]  to  be  assumed  as  the 
standard  both  for  specific  gravity  and  density,  then  will  D^  be  unity. 
and 

'S^  =  l^  =  ^-- (454) 

in  which  S  denotes  the  specific  gravity  of  the  body  whose  density 
is  D ;  and  from  which  we  see,  that  when  specific  gravities  and 
densities  are  referred  to  the  same  substance  as  a  standard,  the 
numbers  wdiich   express   the    one  w^ill    also   express   the    other. 

§275. — Bodies  present  themselves  under  every  variety  of  condi- 
tion—gaseous, liquid,  and  solid  ;  and  in  every  kind  of  shape  and  of 
all  sizes.  The  determination  of  their  specific  gravity,  in  every  in- 
stance, depends  upon  our  ability  to  find  the  weight  of  an  equal 
volume  of  the  standard.  When  a  solid  is  immersed  in  a  fluid,  it 
loses  a  portion  of  its  weight  equal  to  that  of  the  displaced  fluid. 
The  volume  of  the  body  and  that  of  the  displaced  f:uid  are  equal. 
Hence  the  weight  of  the  body  in  vacuo,  divided  by  its  loss  of 
weight  when  immersed,  will  give  the  ratio  of  the  weights  of  equal 
volumes  of  the   body  and  fluid ;   and    if  the    latter    be    taken    as    the 

20 


306  ELEMEXTS     OF     ANALYTICAL    MECHANICS. 

standard,  and  the  loss  of  Aveight  be  made  to  occupy  the  denomi- 
nator, this  ratio  becomes  the  measure  of  the  specific  grav.'.ty  of  the 
body  immersed.  For  this  rea^:on,  and  in  view  of  the  consideration 
tliat  it  may  be  obtained  pure  at  all  times  and  places,  tvater  is 
assumed  as  the  general  standard  of  specific  gravities  and  densities 
for  all  bodies.  Sometimes  the  gases  and  vkpors  are  referred  to 
atmospheric  air,  but  the  specific  gravity  of  the  latter  being  known 
as  referred  to  water,  it  is  very  easy,  as  we  shall  presently  see,  to 
pass  from  the  numbers  which  relate  to  one  standard  to  those  that 
refer    to    the    other. 

§  276. — But  water,  like  all  other  substances,  changes  its  density  with 
its  temperature,  and,  in  consequence,  is  not  an  invariable  standard. 
It  is  hence  necessary  either  to  employ  it  at  a  constant  temperature, 
or  to  have  the  means  of  reducing  the  apparent  specific  gravities,  as 
determined  by  means  of  it  at  different  temperatures,  to  what  they 
would  have  been  if  the  water  had  been  at  the  standard  temperature. 
The  former  is  generally 'impracticable ;    the  latter  is  easy. 

Let  D  denote  the  density  of  any  solid,  and  S  its  specific  gravity, 
as  determined  at  a  standard  temperature  corresponding  to  which  the 
density  of  the  water  is  D^.     Then,  Equation  (453), 

Again,  if  S'  denote  the  specific  gravity  of  the  same  body,  as  indi- 
cated by  the  water  when  at  a  temperature  diflicrent  from  the  stan- 
dard, and  corresponding  to  which  it  has  a  density  D^^,  then  will 

S'-  -^  . 

Dividing  the  first  of  these  equations  by  the  second,  we  have 

S'        D/ 
whence, 

'S'=  '^'•^; (455) 

and   if  the   density  D^  ,  be  taken  as  unity, 

S=  S'-D,,. (450) 


MECHANICS     OF    FLUIDS. 


307 


That  is  to  say,  tlie  sj^ecijic  gravitij  of  a  body  as  determined  at  the 
standard  temperature  of  the  water,  is  equal  to  its  specific  gravity  deter- 
mined at  any  other  temperature,  multiplied  by  the  density  of  the 
water  corresponding  to  this  temperature,  the  density  at  the  standard 
temperature    being  regarded  as  unity. 

To  make  this  rule  practicable,  it  becomes  necessary  to  find  the 
relative  densities  of  water  at  different  temperatures.  For  this  pur- 
pose, take  any  metal,  say  silver,  that  easily  resists  the  chemical 
action  of  water,  and  whose  rate  of  expansion  for  each  degi'ee  of 
Fahr.  thermometer  is  accurately  known  from  experiment;  give  it 
the  form  of  a  slender  cylinder,  that  it  may  readily  conform  to  the 
temperature  of  the  water  when  immersed.  Let  the  length  of  the 
cylinder  at  the  temperature  of  32°  Fahr.  be  denoted  by  /,  and  the 
radius   of  its   base    by    ml;    its  volume    at    this  temperature  will  be 

'n'  m'^P  X   I  —  If  m^  P- 

Let  n  I  be  the  amount  of  expansion  in  length  for  each  degree  of 
the  thermometer  above  32°.  Then,  for  a  temperature  denoted  by 
t,  will    the   whole    expansion   in  length  be 

n  I  X  {t  —  32°), 

P 
and    the    entire  length  of  the  cylin- 
der  will   become 

l+n  I  {t-S2°)^l[l-j-n  {t~S2°)]', 

which,  substituted  for  I  in  the  first 
expression,  will  give  the  volume 
for  the  temperature  t,  equal  to 

*m2/3[i  -^  n{t  —  32°)] 3. 

The  cylinder  is  now  weighed  in 
vacuo  and  in  the  water,  at  differ- 
ent temperatures,  varying  from  32" 
upward,  through  any  desirable  range, 
say  to  one  hundred  degrees.  The 
temperature  at  each  process   being  i^_,,  -,-_  _  ^j.-L-..-) 

substituted  above,  gives  the  volume 
of  the    displaced   fluid ;    the    weight   of    the    displaced   fluid   is    known 


303 


ELEMENTS     OF    ANALYTICAL    MECHANICS. 


from  the  loss  of  weight  of  the  cylinder.  Dividing  this  weight  by 
the  volume,  gives  the  weight  of  the  unit  of  volume  of  the  water  at 
the  temperature  t.  It  was  found  by  Skirnpfer,  that  the  weight  of 
the  unit  of  volume  is  greatest  when  the  temperature  is  38°.75  Fah- 
renheit's scale.  Taking  the  density  of  the  Avater  at  this  temperature 
as  unity,  and  dividing  the  weight  of  the  unit  of  volume  at  each  of 
the  other  temperatures  by  the  weight  of  the  unit  of  volume  at  this, 
88°.75,  Table  11  will  result. 

The  column  under  the  head  V,  will  enable  us  to  determine  how 
much  the  volume  of  any  mass  of  water,  at  a  temperature  t,  exceeds 
that  of  the  same  mass  at  its  maximum  density.  For  this  purpose, 
we  have  but  to  multiply  the  volume  at  the  maximum  density  by 
the    tabular   number   corresponding  to  the  given  temperature. 


§  277. — Before  proceeding  to  the  practical  methods  of  finding  the 
specific  gravity  of  bodies,  and  to  the  variations  in  the  processes 
rendered  necessary  by  the  peculiarities  of  the  different  substances, 
it  will  be  necessary  to  give  some  idea  of  the  best  instruments  em- 
ployed for  this  purpose.  These  are  the  Hydrostatic  Balance  and 
Nicholsoii's  Hydrometer. 

The  first  is  similar  in  principle  and  form  to  the  common  balance. 
It  is  provided  with  numerous 
Aveights,  extending  through  a 
wide  range,  from  a  small 
fraction  of  a  grain  to  several 
ounces.  Attached  to  the  un- 
der surface  of  one  of  the 
basins  is  a  small  hook,  from 
which  may  be  suspended 
any  body  by  means  of  a 
thin  platinum  wire,  horse- 
hair, or  any  other  delicate 
thread  that  will  neither  absorb 
nor  yield  to  the  chemical  ac- 
tion of  the  fluid   in    which   it  may  be   desirable  to  immerse  it. 

Nicholson'' s  Hydrometer  consists  of  a  hollow  metalic  ball  J,  through 


MECHANICS     OF    FLUIDS. 


309 


—/U 


the  centre  of  which  passes  a  metallic  wire,  prolonged  in  both  di- 
rections beyond  the  surface,  and  supporting 
at  either  end  a  basin  B  and  B' .  The 
cuiioavitics  of  these  basins  are  turned  in 
the  same  direction,  and  the  basin  B'  is 
made  so  heavy  that  when  the  insti'ument 
is  placed  in  water  the  stem  C  C  shall  be 
vertical,  and  a  weight  of  500  grains  being 
placed  in  the  basin  B,  the  whole  instrument 
will  sinlv  till  the  upper  surface  of  distilled 
water,  at  the  standard  temperature,  comes  to 
a    jioiut    C  marked    on    the  upper  stem  near 

its    middle.     This    instrument    is    provided     with     weights    similar    to 
those   of  the    Hydrostatic   Balance. 

§278. — (1).  If  the  body  he  solid,  insoluble  in  water,  and,  xo'dl  sink 
in  thai  fluid,  attach  it,  by  means  of  a  hair,  to  the  hoolc  of  the 
basin  of  the  hydrostatic  balance  ;  counterpoise  it  by  j^lacing  weights 
in  the  opposite  scale  ;  now  immerse  the  boi;ly  in  water,  and  restore 
the  equilibrium  by  placing  weights  in  the  basin  above  the  body, 
and  note  the  temperature  of  the  water.  Divide  the  weights  in  the 
basin  to  which  the  body  is  not  attached  by  those  in  the  basin  to 
which  it  is,  and  multiply  the  cpiotient  by  the  density  corresponding 
to  the  temperature  of  the  water,  as  given  by  the  table ;  the  result 
will   be   the    specific  gravity. 

Thus  denote  the  specific  gravity  by  ^S',  the  density  of  the  water 
by  D^^ ,  the  weight  in  the  first  case  by  W,  and  that  in  the  scale 
above    the  solid  by  iv,  then  will 

S  =  D,.  X  — . 


(2).  If  the  body  be  insoluble,  but  ivill  not  sink  in  loater,  as  would 
be  the  case  with  most  varieties  of  wood,  wax,  and  the  like,  attach 
to  it  some  body,  as  a  metal,  whose  weight  in  the  air  and  loss  of 
weight  in  the  water  are  previously  found.  Then  proceed,  as  in  the 
case  before,  to  find  the  weights  which  will  counterpoise  the  com- 
pound   in    air  and   restore   the    equilibrium  of  the   balance  when  it  is 


310  ELEMENTS     OF    ANALYTICAL    MECHANICS. 

immersed  in  the  water.  From  the  weight  of  tlie  compound  in  air, 
subtract  that  of  the  denser  body  in  air ;  ftoni  the  loss  of  weio-ht 
of  the  compound  in  water,  subtract  that  of  the  denser  body  ; 
divide  the  first  difference  by  the  second,  and  multiply  by  the  density 
of  the  water  answering  to  its  temperature,  and  the  result  will  be 
the    specific   gravity  sought. 

Exain2)le. 

grs.  

A  piece  of  wax  and  copper  in  air  =  438      =  TF+  W, 

Lost   on    immersion  in  water  -     -  =     95,8  =   w  +   xv\ 

Copper  in  air —  388      =  IF', 

Loss  of  copper  in  water      -     •     -  =    44,2  =  lu'. 

Then 

W  +  W  -  W  =  438  -  388    =  50,     ^  IF", 
w   +   xu'  —  %o'   ^  95,8  -  44,3  =  51,6  =  w. 

Temperature  of  water  43°,25, 

J)^,  =  0,999952, 

W  50 

S  =  I)^,  X  —    =  0,999952  X    — —  =  0,908. 
■w  5l,b 

(3).  If'  tJte  body  rcadil//  dissolve  in  ivaier,  as  many  of  the  salts, 
sugar,  tvc.,  find  its  apparent  specific  gravity  in  some  liquid  in  which 
it  is  insoluble,  and  multiply  this  apparent  specific  gravity  by  the 
density  or  specific  gravity  of  the  liquid  referred  to  water  at  its 
maximum  density  as  a  standard ;  the  product  will  be  the  true  specific 
gravity. 

If  it  be  inconvenient  to  provide  a  liquid  in  which  the  solid  is 
insoluble,  saturate  the  water  with  the  substance,  and  find  the  appa- 
rent specific  gravity  with  the  water  thus  saturated.  Multiply  this 
apparent  specific  gravity  by  the  density  of  the  saturated  fiiiid,  and 
the  product  will  l)e  the  specific  gravity  referred  to  the  standard. 
This  is  a  common  method  of  finding  the  specific  gravity  of  gunpow- 
der, the  water    being    saturated  with  nitre. 

(4).  //'  the  hndij  be  a  liquid,  select  some  solid  that  will  i-esist  its 
chemical    action,    as    a    massive    piece    of    glass    suspended    from    fine 


MECHANICS     OF     FLUIDS.  311 

platinum  wire ;  weigh  it  in  air,  then  in  water,  and  fmally  in  the 
liquid ;  the  difterences  between  the  first  weight  and  each  of  the 
latter,  will  give  the  weights  of  equal  volumes  of  water  and  the 
liquid.  Divide  the  weight  of  the  liquid  by  that  of  the  water,  and 
the  quotient  will  be  the  specific  gravity  of  the  liquid,  provided  the 
temperature  of  water  be  at  the  standard.  If  the  water  have  not 
the  standard  temperature,  multiply  this  apparent  specific  gravity  by 
the  tabular  density  of  the  water  corresponding  to  the  actual  tem- 
perature. 

Exam])le. 

grs. 

Loss  of  glass  in  water  at  41°,   150      =  w\ 
"  "  sulphuric  acid,  277,5  =  lo, 

277  5 
S  =  -— ^  X  0,999988  =  1,85. 
loO 

(5).  If  the  body  be  a  gas  or  vajmr,  provide  a  large  glass  flask- 
shaped  vessel,  weigh  it  when  filled  with  the  gas  ;  withdraw  the  gas, 
which  may  be  done  by  means  to  be  explained  presently,  fill  with 
water,  and  weigh  again  ;  finally,  withdraw  tlie  water  and  exclude  the 
air,  and  weigh  again.  This  last  weight  subtracted  from  the  first, 
will  give  the  weight  of  the  gas  that  filled  the  vessel,  and  subtracted 
from  the  second  will  give  the  weight  of  an  equal  volume  of  water; 
divide  the  weight  of  the  gas  by  that  of  the  water,  and  multiply 
by  the  tabular  density  of  the  water  answering  to  the  actual  tem- 
perature of  the  latter ;  the  result  will  be  the  sijecific  gravity  of 
the    gas. 

The  atmosphere  in  which  all  these  operations  must  be  performed, 
varies  at  diflfercnt  times,  even  during  the  same  day,  in  respect  to 
temperature,  the  weight  of  its  column  which  presses  upon  the  earth, 
and  the  quantity  of  moisture  or  aqueous  vapor  it  contains.  That  is 
to  say,  its  density  depends  upon  the  state  of  the  thermometer,  barom- 
eter, and  hygrometer.  On  all  these  accounts  corrections  must  be 
made,-  before  the  specific  gravity  of  atmospheric  aij*,  or  that  of  any 
gas  exposed  to  its  pressure,  can  be  accurately  determined.  The  prin- 
ciples according  to  which  these  corrections  are  made,  will  be  discussed 
when  we  i!ome  to  treat  of  the  properties  of  elastic   fluids. 


312 


ELEMENTS  OF  ANALYTICAL  MECHANICS. 


To  find  the  specific  gravity  of  a  solid  by  means  of  Nicholson's 
Hydrometer,place  the  instrument  in  water,  and  add  weights  to  the 
upper  basin  until  it  sinks  to  the  mark  on  the  upper  stem;  rem.ove 
the  weights  and  i)]ace  the  solid  in  the  upper  basin,  and  add  weights 
till  the  hydrometer  sinks  to  the  same  point ;  the  difference  between 
the  first  weights  and  those  added  with  the  body,  will  give  the 
weight  of  the  latter  in  air.  Take  the  body  from  the  upper  basin, 
leaving  the  Aveights  behind,  and  place  it  in  the  lower  basin ;  add 
weights  to  the  upper  basin  till  the  instrument  sinks  to  the  same  point 
as  before,  the  last  added  weights  will  be  the  weight  of  the  water 
displaced  by  the  body  ;  divide  the  weight  in  air  by  the  weight  of 
the  displaced  water,  and  multiply  the  quotient  by  the  tabular  density 
of  the  water  answering  to  its  actual  temperature  ;  the  result  Avill  be 
the  specific  gravity  of  the  solid. 

To  find  the  specific  gravity  of  a  fluid  by  this  instrument,  immerse 
it  in  water  as  before,  and  by  weights  in  the  upper  basin  sink  it  to 
the  mark  on  the  upper  stem ;  add  the  weights  in  the  basin  to  the 
weight  of  the  instrument,  the  sum  will  be  the  weight  of  the  dis- 
placed  water.  Place  the  instrument  in  the  fluid  whose  specific  gravity 
is  to  be  found,  and  add  weights  in  the  upper  basin  till  it  sinks  to 
the  mark  as  before;  add  these  weights  to  the  M'eight  of  the  instru- 
ment, the  sum  will  be  the  weight  of  an  equal  volume  of  the  fluid ; 
divide  this  weight  by  the  weight  of  the 
water,  and  multiply  by  the  tabular  density 
corresponding  to  the  temperature  of  the 
water,  the  result  will  be  the  specific  gravity. 

§  279. — Besides  the  hydrometer  of  Nichol- 
son, which  requires  the  use  of  weights,  there 
is  another  form  of  this  instrument  which  is 
employed  solely  in  the  determination  of  the 
specific  gravities  of  liquids,  and  its  indications 
arc  given  by  means  of  a  scale  of  equal  parts. 
It  is  called  the  Scale-Areometer.  It  consists, 
generally,  of  a  glass  vial-shaped  vessel  A,  ter- 
minating at  one  end  in  a  long  slender  neck  C, 
to  receive  the  scale,  and   at   the    other    in   a 


MECHAISriCS     OF     FLUIDS.  313 

small  globe  i?,  filled,  with  some  heavy  substance,  as  lead  or  mercury 
to  keep  it  upright  when  immersed  in  a  fluid.  The  application  and 
use  of  the  scale  depend  upon  this,  that  a  body  floating  on  the  surface 
of  different  liquids,  will  sink  deeper  and  deeper,  in  proportion  as  the 
density  of  the  fluid  approaches  that  of  the  body  ;  for  when  the  body 
is  at  rest  its  weight  and  that  of  the  displaced  fluid  must  be  equal. 
Denoting  the  volume  of  the  instrument  by  V,  that  of  the  dis- 
placed fluid  by  F',  the  density  of  the  instrument  by  D,  and  that 
of  the    fluid   by    D',  we    must   always    have 

gVD  =:r/V'D'; 

in  which  </  denotes  the  force  of  gravity,  the  first  member  the  wei^iit 
of  the  instrument,  and  the  second  that  of  the  displaced  fluid.  Divi- 
ding both  members  by  £>'  V,  and  omitting  the  common  factor  ff 
we    have 

D    _   V 

In  which,  if  the  densities  be  equal,  the  volumes  must  be  equal ; 
if  the  density  B'  of  the  fluid  be  greater  than  D,  or  that  of  the 
solid,  the  volume  F  of  the  solid  must  be  greater  than  F',  or  that 
of  the  displaced  fluid ;  and  in  proportion  as  B'  increases  in  respect 
to  D,  will  F'  diminish  in  respect  to  F;  that  is,  the  solid  will 
rise  higher  and  higher  out  of  the  fluid  in  proportion  as  the  den- 
sity of  the  latter  is  increased,  and  the  reverse.  The  neck  C  of 
the  vessel  should  be  of  the  same  diameter  throughout.  To  estab- 
lish the  scale,  the  instrument  is  placed  in  distilled  water  at  the 
standard  temperature,  and  when  at  rest  the  place  of  the  surflice 
of  the  water  on  the  neck  is  marked  and  numbered  1  ;  tlie  in^itru- 
ment  is  then  placed  in  some  heavy  solution  of  salt,  whose  specific 
gravity  is  accurately  known  by  means  of  the  Hydrostatic  Balance, 
and  when  at  rest  the  place  on  the  neck  of  the  fluid  surface  is  again 
marked  and  characterized  by  its  appropriate  number.  The  same  pro- 
cess being  repeated  for  rectified  alcohol,  will  give  another  point 
towards  the  opposite  extreme  of  the  scale,  which  may  be  completed 
by   graduation. 


814         ELEMENTS     OF     ANALYTICAL     MECHANICS. 

To  use  this  instrument,  it  will  be  sufficient  to  immerse  it  in  a 
fluid  and  take  the  number  on  the  scale  which  coincides  with  the 
surface. 

To  ascertain  the  circumstances  which  determine  the  sensibility 
both  of  the  Scale-Areometer  and  Nicholson's  Hydrometer,  let  s  de- 
note the  specific  gravity  of  the  fluid,  c  the  volume  of  the  vial,  I  the 
length  of  the  immersed  portion  of  the  narrow  neck,  r  its  semi-diame- 
ter, and  'w  the  total  weight  of  the  instrument.  Then  will  tt  r^,  denote 
the  area  of  a  section  of  the  neck,  and  *  r^  /,  the  volume  of  fluid  dis- 
placed by  the  immersed  part  of  the  neck.  The  weight,  therefore,  of 
the  whole  fluid  displaced  by  the  vial  and  neck  will  be 

s  c  -\-  s  T!'  i'"^  I ; 

but  this  must  be  equal  to  the  weight  of  the  instrument,  whence, 

^v  z=:  s  (c  +  ir  r"^  I), 
fi'om  which  we  deduce. 


c  +  irr^r 
1  =  "-^^. .(457) 

Now,  immersing  the  instrument  in  a  second  fluid  Avhose  specific  gravi- 
ty is  &•',  the  neck  will  sink  through  a  distance  l\  and  from  the  last 
equation  we  have 

J,  _w  -  s'c^ 

subtracting  this  equation  from  that  above  and  reducing,  we  find 

If  r^  \    s  A'     / 

The  diflerence  I  —  I'  is  the  distance  between  two  points  on  the  scale 
which  indicates  the  diflerence  s'  —  s  of  specific  gravities,  and  this 
we  see  becomes  longer,  and  the  instrument  more  sensible,  therefore, 
in  proportion  as  w  is  made  greater  and  r  less.  Whence  we  con- 
clude that  tlie  Areometer  is  the  more  valuable  in  proportion  as  the 
vial  portion  is  made  larger  and  the  neck  smaller. 


MECHANICS     OF    FLUIDS.  315 

»f  the  specific  gravity  of  the  fluid  remain  the  same,  which  is  the 
case  -with  Nicholson's  Hydrometer,  and  it  becomes  a  question  to 
know  the  effect  of  a  small  weight  added  to  the  instrument,  denote 
this   weight  by  w\  then  will  Equation  (457)  become 

IV  -]-  'w'  —  s  c 

If  7'^  S  ' 

subtracting  from  this  Equation  (457),  we  find 


It  T"  S 


Prom  which  we  see  that  the  narrower  the  upper  stem  of  Nicholson's 
iustrument,  the  greater  its  sensibility. 

The  knowledge  of  the  specific  gravities  or  densities  of  different 
substances,  Table  III,  is  of  great  importance,  not  only  for  scientific 
purposes,  but  also  for  its  application  to  many  of  the  useful  arts. 
This  knowledge  enables  us  to  solve  such  problems  as  the  follow 
ing,  viz.  : — 

1st.  The  weight  of  any  substance  may  be  calculated,  if  its  volume 
and    specific   gravity  be    known. 

2d.  The  volume  of  any  body  may  be  deduced  from  its  specific 
gravity  and  weight.     Thus  we   have   always 

W  =  ffDV; 

in  -which  r/  is  the  force  of  gravity,  D  the  density,  V  the  volun^e, 
and  W  the  weight,  of  which  the  unit  of  measure  is  the  weight  of 
a   unit    of  volume    of  water  at    its    maximum    density. 

Making  D  and    V  equal    to    unity,  this  equation  becomes 

but  if  the  density  be  one,  the  substance  must  be  water  at  38°,75 
Eahr.  The  weight  of  a  cubic  foot  of  water  at  60°  is  02,5  lbs.,  and, 
therefore,  at  38°,75,  it  is 

lbs. 

^^'^  62,556 ; 


0,99914 

whence,  if  the   volume   be  expressed   in    cubit   feet. 


lbs. 

W  =  62,556  X  J)V (458) 


ne 


ELEMENTS  OF  ANALYTICAL  MECHANICS. 


in  which    IF  is  expressed  in    pounds ;    and   if  the   unit    of  vohinic  be 
a   cubic   inch, 


Also, 


f,0  KKC 

W  =     ^1 ,      DV=:  0,036201  D  V, 

1726  ' 


V  = 


V.  = 


w. 


.  lbs. 

62,556  .  Z> 


W. 


/bs. 

0,036201  .  I) 


(459) 

(460) 

(461) 


Example  1. — Recpiired  the  weight  of  a  block  of  dry  fir,  containing 
50    cubic  inches.     The    specific  gravity  or  density  of  dry  fir  is  0,555, 

and    V  —  bO  \    substituting   these    values  in  Ecpiation  (459), 

lbs. 

W  =  0,030201  X  0,555  X  50  ==  L00457. 

Exarivph  2. — How  many  cuhic  inches  are  there  in  a  12-pound 
cannon-ball  1  Here  W  is  12  pounds,  the  mean  specific  gravity  of 
cast   iron  is  7,251,  which,  in  Equation  (461),  give 

12 


0,036201   X  7,251 


r=  45,6. 


AT^klOSPHEKIC    PKESSUEE. 

§280. — The  atmosphere  encases,  as  it  were,  the  whole  earth.  It 
has  weight,  else  the  repulsive  action  among  its  own  particles  wolild 
cause  it  to  expand  and  extend  itself  through  space.  The  weight  of 
the  upper  stratum  of  the  atmosphere  is  in  equilibrio  with  the  re- 
pidsive  action  of  the  strata  below  it,  and  this  condition  determines 
the    exterior  limit. 

Since  the  atmosphere  has  weight,  it  must 
exert  a  pressure  upon  all  bodies  within  it. 
To  illustrate,  fill  with  mercury  a  glass  tube, 
about  32  or  33  inches  long,  and  closed  at 
one  end  by  an  iron  stop-cock.  Close  the 
open  end  by  pressing  the  finger  against  it, 
and  invert  the  tube  in  a  basin  of  mercury ; 
remove  the  finger,  the  mercury  will  not 
escape,  but  remain    a])parently   suspended,    at 


MECHANICS     OF    FLUIDS.  31T 

the  le\'el  of  the  ocean,  nearly  30  inches  above  the  sui-flxce  of  the 
mercury  in  the    basin. 

The  atmospheric  air  presses  on  tlie  mercury  with  a  force  sufficient 
to  maintain  the  quiclvsilver  in  the  tube  at  a  height  of  nearly  30 
inches  ;  ■whence,  the  intensitij  of  its  i^ressure  must  he  equal  to  the  xoeighi 
of  a  column  of  mercury  whose  base  is  equal  to  that  of  the  surface 
pressed  and  whose  altitude  is  about  30  inches.  The  force  thus  exerted, 
is  called  the  atmospheric  pressure. 

The  absolute  amount  of  atmospheric  pressure  was  first  discovered 
by  Torricelli,  and  the  tubes  employed  in  such  experiments  are  called, 
on  this  account,  Torricellian  tubes,  and  the  vacant  space  above  the 
mercury  in    the   tube,  is  called    the   Torricellian  vacuum. 

The  pressure  of  the  atmosphere  at  the  level  of  the  sea,  support- 
ing as  it  does  a  column  of  mercury  30  inches  high,  if  we  suppose 
the  bore  of  the  tube  to  have  a  cross-section  of  one  square  inch, 
the  atmospheric  pressure  up  the  tube  will  be  exerted  upon  this 
extent  of  surfiice,  and  will  support  30  cubic  inches  of  mercury. 
Each  cubic  inch  of  mercury  weighs  0,49  of  a  pound — say  half  a 
pound — from  which  it  is  apparent  that  the  surfaces  of  all  bodies,  at 
the  level  of  the  sea,  are  subjected  to  an  atmospheric  pressure  of  fifteen 
pounds   to  each  square  inch. 

BAKOMETEE. 

§281. — Tlie  atmosphere  being  a  heavy  and  elastic  fluid,  is  com- 
pressed by  its  own  weight.  Its  density  cannot  be  the  same  through- 
out, but  diminishes  as  we  approach  its  upper  limit  where  it  is  least, 
being  greatest  at  the  surfiice  of  the  earth.  If  a  vessel  filled  M'ith 
air  be  closed  at  the  base  of  a  high  mountain  and  afterwards  opened 
on  its  summit,  the  air  will  rush  out ;  and  the  vessel  being  closed 
again  on  the  summit  and  opened  at  the  base  of  the  mountain,  the 
air  will  rush  in. 

The  evaporation  which  takes  place  from  large  bodies  of  water, 
the  activity  of  vegetable  and  animal  life,  as  well  as  vegetable  decom- 
positions, throw  considerable  quantities  of  aqueous  vapor,  carbonic 
acid,  and    other    foreign    ingredients   temporarily    into    the    permanent 


31 S 


ELEMENTS  OF  ANALYTICAL  MECHANICS. 


portions  of  the  atmosphere.  These,  together  with  its  ever-varying 
temperature,  keep  the  density  and  elastic  force  of  the  air  in  a 
state  of  almost  incessant  change.  These  changes  are  indicated  by 
the  Barometer^  an  instrument  employed  to  measure  the  intensity  of 
atmospheric  pressure,  and  frequently  called  a  weather-glass,  because 
of  certain  agreements  found  to  exist  between  its  indications  and  the 
state    of  the    weather. 

Tlie  barometer  consists  of  a  glass  tube  about  thirty-four  or  thirty- 
live  inches  long,  open  at  one  end,  partly  filled  with  distilled  mer- 
cury, and  inverted  in  a  small  cistern  also  containing  mercury.  A 
scale  of  equal  parts  is  cut  upon  a  slip  of  metal,  and  placed  against 
the  tube  to  measure  the  height  of  the  mercurial  column,  the  zero 
being  on  a  level  with  the  surface  of  the  mercury  in  the  cistern. 
The  elastic  force  of  the  air  acting  freely  upon  the  mercury  in  the 
cistern,  its  pressure  is  transmitted  to  the  interior  of  the  tube,  and 
sustains  a  column  of  mercury  whose  weight  it  is  just  sufficient  fco 
counterbalance.  If  the  density  and  consequent  elastic 
force  of  the  air  be  increased,  the  column  of  mercury 
will  rise  till  it  attain  a  corresponding  increase  of 
weight;  if,  on  the  contrary,  the  density  of  the  air 
diminish,  the  column  will  fall  till  its  diminished 
■weight  is   sufficient   to   restore    the    equilibrium. 

In  the  Common  Barometer,  the  tube  and  its  cis- 
tern are  partly  inclosed  in  a  metallic  case,  upon 
which  the  scale  is  cut,  the  cistern,  in  tliis  case,  bav- 
in"- a  flexible  bottom  of  leather,  against  which  a 
plate  a  at  the  end  of  a  screw  b  is  made  to  press, 
in  order  to  elevate  or  depress  the  mercury  in  the 
cistern  to  the  zero   of  the   scale. 

De  Luc's  Sijyhon  Barometer  consists  of  a  glass 
tube  bent  upward  so  as  to  form  two  unequal  par- 
allel legs :  the  longer  is  hermetically  sealed,  and 
constitutes  the  Torricellian  tube ;  the  shorter  is  open, 
and   on    the    surfoce  of  the   quicksilver   the   pressure  c^b 

of    the    atmosphere   is    exerted.      The    difference    be- 
tween   the   levels    in    the  longer   and   shorter   legs   is    the   barometric 


MECHANICS     OF     FLUIDS. 


319 


30^ 
29 


height.     The    most  CDiivenient  and  practicable  way  of  measiu-ing  this 
difference,    is   to    adjust    a    movable    scale    between 
the    two   legs,    so    that   its    zero    may    be    made    to 
coincide    with    the    level     of    the    mercury    in     the 
shorter   leg. 

Different  contrivances  have  been  adopted  to  ren- 
der the  minute  variations  in  the  atmospheric  pres- 
sure, and  consequently  in  the  height  of  the  barome- 
ter, more  readily  perceptible  by  enlarging  the  di- 
visions on  the  scale,  all  of  which  devices  tend  to 
hinder  the  exact  measurement  of  the  length  of  the 
column.  Of  these  we  may  name  Morland's  Diago- 
nal, and  Hook's  Wheel-Barometer,  but  especially 
Huygen's  Double-Barometer. 

The    essential    properties    of    a   good    barometer 

are     Avidth  of   tube ;    purity    of  the  mercury  ;    accu-  

rate  graduation    of  the    scale ;  and   a   good   vernier. 

§  282. — The  barometer  may  be  used  not  only  to  measure  the 
pressure  of  the  external  air,  but  also  to  determine  the  density  and 
elasticity  of  peht-up  gases  and  vapors.  When  thus  employed,  it  is 
called  the  barometer-gavge.  In  every  case  it  Will 
only  be  necessary  to  establish  a  free  connection 
between  the  cistern  of  the  baroziieter  and  the  vessel 
containing  the  fluid  whose  elasticity  is  to  be  indi- 
cated ;  the  height  of  the  mercury  in  the  tube, 
expressed  in  inches,  reduced  to  a  standard  tempera- 
ture, and  multiplied  by  the  known  weight  of  a 
cubic  inch  of  mercury  at  that  temperature,  Avill 
give  the  pressure  in  pounds  on  each  square  inch. 
In  the  case  of  the  steam  in  the  boiler  of  an  en- 
gine, the  upper  end  of  the  tube  is  sometimes  left 
open.  The  cistern  A  is  a  steam-tight  vessel,  partly 
fdled  with  mercury,  a  is  a  tube  communicating 
V/ith  the  boiler,  and  through  which  the  steam  flows 
and  presses  upon  the  mercury  ;  the  barometer  tube 
6  c,    opiMi    at    top,    reaches    nearly    tu    the    bottom    of    the    vessel    A. 


320 


ELEMENTS     OF     ANALYTICAL     IrlECHANICS. 


liMvmg  attached  to  it  a  scale  whose  zero  coincides  with  the  level 
of  the  quicksilver.  On  the  ri^ht  is  marked  a  scale  of  inches,  and 
on   the   left    a    scale    of  atmospheres. 

If  a  very  high  pressure  were  exerted,  one  of  several  atmospheres 
for  example,  an  apparatus  thus  constructed  would 
require  a  tube  of  great  length,  in  Avhieh  case  Ma- 
riotfr's  manometer  is  considered  preferable.  The  tube 
being  filled  with  air  and  the  upper  end  closed,  the 
surface  of  the  mercury  in  both  branches  will  stand 
at  the  same  level  as  long  as  no  steam  is  admitted. 
The  steam  being  admitted  through  d,  presses  on  the 
surface  of  the  mercury  a  and  forces  it  up  the  branch 
be,  and  the  scale  from  i  to  c  marks  the  force  of 
compression  in  atmospheres.  The  greater  width  of 
tube  is  given  at  o,  in  order  that  the  level  of  the 
mercury  at  this  point  may  not  be  materially  affected 
by  its  ascent  up  the  branch  be,  the  point  a  being  the  zero  of  the 
scale. 


§283. — Another  very  important  use  of  the  barometer,  is  to  find 
the  difference  of  level  between  two  places  on  the  earth's  surface,  as 
the  foot    and    top    of  a    hill    or    mountain. 

Since  the  altitude  of  the  barometer  depends  on  the  pressure  of 
the  atmosphere,  and  as  this  force  depends  upon  the  height  of  the 
pressing  column,  a  shorter  column  will  exert  a  less  pressure  than  a 
longer  one.  The  quicksilver  in  the  barometer  falls  when  the  instru- 
ment is  carried  from  the  foot  to  the  top  of  a  mountain,  and  rises 
again  when  restored  to  its  first  position :  if  taken  down  the  shaft 
of  a  mine,  the  barometric  column  rises  to  a  still  greater  height.  At 
the  foot  of  the  mountain  the  whole  column  of  the  atmosphere,  from 
its  utmost  limits,  j)resses  with  its  entire  weight  on  the  mercury; 
at  the  top  of  the  mountain  this  weight  is  diminished  by  that  of 
the  intervening  stratum  between  the  two  stations,  and  a  shorter 
column  of  mercury   will    be    sustained    by    it. 

It  is  well  known  that  the  surface  of  the  earth  is  not  uniform, 
and  does  not,  in  consequence,  sustain   an  equal  atmospheric   pressure 


MECHANICS     OF    FLUIDS.  321 

at  its  different  points ;  whence  tlie  mean  altitude  of  the  barometric 
column  will  vary  at  different  places.  This  furnishes  one  of  the 
best  and  most  expeditious  means  of  getting  a  profile  of  an  extended 
section  of  the  earth's  surliice,  and  makes  the  barometer  an  instru- 
ment of  great  value  in  the  hands  of  the  traveller  in  search  of 
geographical   information. 

§  284. — To  find  the  relation  which  subsists  between  the  altitudes 
of  two  barometric  columns,  and  the  difference  of  level  of  the  points 
where  they  exist,  resume  Equation  (-427).  The  only  extraneous  force 
acting  being  that  of  gravity,  we  have,  taking  the  axis  z  vertical, 
and  counting  z   positive   upwards, 

X=0;     FnzO;     Z=-g. 
and   hence, 

p  =  Ce~^f (462) 

Making  z  =  0,  and  denoting  the  corresponding  pressure  by,  j:)^,  we  find 

anil   dividing  the  last    equation  by  this  one,  "1^  i 

p         -^-1   '  Oi  ?-  ■-    '''■■■<■    b: 

whence,  denoting   the  reciprocal  of  the   common  modulus  by  M, 

.  =  ^-Iogi^ (463) 

9  P 

Denote  by  h^  and  /i,  the  barometric  heights  at  the  lower  and  upper 
stations,  respectively,  then  will 

II  -   h. 

p  A' 

and  reducing  the  barometric  column  h  to  what  it  would  have  been 
had  the  temperature  of  the  mercury  at  the  upper  not  differed  from 
that   at    the   lower  station,  by  Equation  (394),  we  have 

Pj_  _  ^^ . 

p    ~  A  [1  +(2^- T').  0,0001001]" 

in  which   T  denotes  the  temperature  of  the  mercury  at  '.he  lowfr  and 

T'  that  at  the  upper  station. 

21 


823  ELEMENTS     OF    ANALYTICAL    MECHANICS. 

Moreovei',  Equation  (381), 

ff  =  c/'  {1  —  0,002551  cos  2  4.) ; 

in  which, 

/ 
ff'  =  32,1808  ==  force   of  gravity  at  the  latitude  of  45°. 

P 
Substituting   the  value  of  —  • ,    of  y,  and  that  of  P,  as  given    b^; 

Equation  (393),  in  Equation   (463),  we  find 

_  MJ)Ji.,,    1  + (^-32^)0,00204  r/i^ 1_ 1 

^~       B^'    *1-O,002551cos24.^  °^L/i  ^  l  +  (y'- r')0,0001001  J" 

In  this  it  will  be  renaembered  that  t  denotes  the  temperature  of 
the  air;  but  this  may  not  be,  indeed  scarcely  ever  is,  the  same  at 
both  stations,  and  thence  arises  a  difficulty  in  applying  the  fonnula. 
But  if  we  represent,  for  a  moment,  the  entire  factor  of  the  second 
member,  into  which  the  factor  involving  t  is  multiplied,  by  A",  then 
we   may  write 

z  =z  [\  +  [t  -  32°)0,00204]  X. 

If  the  temperature  of  the  lower  station  be  denoted  by  t^ ,  and  this 
temperature  be  the  same  throughout  to  the  upper  station,  then  will 

z,r=\l^  {t,  -  32°)  0,00204]  X. 

And  if  the  actual  temperature  of  the  ^i-pper  station  be  denoted  by  t' , 
and  this  be  supposed  to  extend  to  the  lower  station,  then   would 

£'  =  [1  +  {f  -  32°)  0,00204]  X. 

Now  if  t^  be  greater  than  t\  which  is  usually  the  case,  then  will  the 
barometric  column,  or  h,  at  the  upper  station,  be  greater  than  would 
result  from  the  temperature  t',  since  the  air  being  more  expanded, 
a  portion  which  is  actually  below  would  pass  above  the  upper 
station  and  press  upon  the  mercury  in  the  cistern  ;  and  because  h 
enters  the  denominator  of  the  value  X,  z,  would  be  too  small. 
Again,  by  supposing  the  temperature  the  same  as  that  at  the  uppei 
station  throughout,  then  would  the  air  be  more  condensed  at  the 
lower  station,  a  portion  of  the  air  would  sink  below  the  upper 
station  that  before  was  above  it,  and  would  cease  to  act  upon  the 
tnercurial  column  /t,  which  would,  in  consequence,  become  too  small ; 


MECHANICS    OF    FLUIDS.  323 

and  this  would   make  z'  too   great.     Taking  a  mean  between  z^  and 
z'  as  the  true   value,  we  find 

2  =  ^'  "^  "^    =  [1  4-  1  (;^  +  /'  _  64°)   0,00204]  X. 


Replacing  X  by  its  value, 


X  log    -/  X 


MDJi,^  l+^(^  +  ^'-64°)0,00204  ^^,_[h,  ^^  1 

1-0,00 


G 


~     D,  1-0,002551  cos  2 -vj^  ^L/i      l  +  (7'-r')0,000100lJ 

ihe  factor  ~ ^5  we  have  seen,  is  constant,  and  it  only  re- 
mains to  determine  its  value.  For  this  purpose,  measure  with 
accuracy  the  difference  of  level  between  two  stations,  one  at  the 
base  and  the  other  on  the  summit  of  some  lofty  mountain,  by 
means  of  a  Theodolite,  or  levelling  instrument — this  Avill  give  the 
value  of  z ;  observe  the  barometric  column  at  both  stations — this 
will  give  h  and  li,  ;  take  also  the  temperature  of  the  mercury  at 
the  two  stations — this  will  give  T  and  T' ;  and  by  a  detached 
thermometer  in  the  shade,  at  both  stations,  find  the  values  of 
1^  and  t! .  These,  and  the  latitude  of  the  place,  being  substituted  in 
the  formula,  every  thing  will  be  known  except  the  co-efficient  in 
question,  which  may,  therefore,  be  found  by  the  solution  of  a  simple 
equation.     In    this   Avay,  it   is    found   that 

^^^"  ^'"    =  60345,51   English  feet ; 

which  will  finally  give  for  s, 

_   ^   r^'-      l  +  K'?,  +  <'-64°)0,00204  Vh, 1 "1 

2_G0o4.x51.-   ^  -_  (^  ()()o.5j  cos"24r>^'''^Lr-"^l  +  (7^-y')0,000100Tj 

To  find  the  difference  of  level  between  any  two  stations,  the  lati- 
tude of  the  locality  must  be  known  ;  it  will  then  only  be  necessary 
to  note  the  barometric  columns,  the  temperature  of  the  mercury, 
and  that  of  the  air  at  the  two  stations,  and  to  substitute  these 
observed    elements    in    this    formula. 

Much  labor  is,  however,  saved  by  the  use  of  a  table  for  the 
computation  of  these  results,  and  we  now  proceed  to  explain  how  it 
may  be  formed  and  used. 


324  ELEMENTS     OF     ANALYTICAL    MECHANICS. 

Make 

60345,51  [1  +  {t^  +  t'  -G4°)  0,001 02]  =  A, 


Then  will 


1  —  0,002551  cos  2 -4. 

1 
1  -{-  {T  -  T)  0,0001 

z  =  AB  -los. 


^  C. 


h 
z^AB.\\og  (7+ log/.,  -log/.]; 

and    taking    the    logarithms  of  both    members, 

log  z  =  log  A  +  log  i?  +  log  [log  C  +  log  h^  —  log  K]  •  .  (404) 

Making  t^  +  t'  to  vary  from  40='  to  102°,  which  will  be  sufficient 
for  all  practical  purposes,  the  logarithms  of  the  corresponding  values 
of  A  are  entered  in  a  column,  under  the  head  A^  opposite  the 
values  t^  +  i\  as    an    argument. 

Causing  the  latitude  4^  to  vary  from  0°  to  90°,  the  logarithms 
of  the  coi'responding  values  of  B  arc  entered  in  a  column  headed 
B.  opposite   the   values   of  4'. 

The  value  of  T  —  T'  being  made,  in  like  manner,  to  vary 
from  —  30°  to  +  30°,  the  logarithms  of  the  corresponding  values 
of  C  are  entered  under  the  head  of  6',  and  opposite  the  values  of 
T  —  T' .  In  this  way  a  table  is  easily  constructed.  Table  IV  was 
computed  by  Samuel  Howlet,  Esq.,  from  the  formula  of  Mr.  Francis 
Bailv,  which  is  very  nearly  the  same  as  that  just  described,  there 
being   but   a   trifling  difference    in    the    co-efficients. 

Takiufif  Equation  (404)  in  connection  wdth  Table  IV,  we  have  this 
rule  for   finding  tlie    alt'tude   of  one    station   above   another,  viz.  :^ 

Take  the  iDjurithm  of  the  barometric  reading  at  the  loiver  station, 
to  w/iich  add  the  number  in  the  column  headed  (7,  opposite  the  ob- 
served value  of  T  —  T\  and  subtract  from,  this  sum  the  logarithm 
of  the  barometric  reading  at  the  upper  station ;  take  the  logarithm 
of  this  difference,  to  tvhich  add  the  mnnbcrs  in  the  colamns  luaJed 
A  and  B,  corresponding  to  the  observed  values  of  t^  +  t'  and  --j/  ; 
the  sum  will  be    the    togaritJjn    of  the   height    in    English  feet. 


MECHANICS    OF    FLUIDS.  6-2'J 

Example. — At   the    mountaiu   of  Guana.xuato,  in  Mexico,  M.   niiui- 
boldt  observed   at   the 

Upper  Station.  flower  Station. 

Detached    thermometer,    i'   =  70*^,4  ;      t^   =  77°, 6. 
Attached  "  r  =  70.4 ;       T  =r  77,G. 

Barometric  column,  h   —  23,00  ;      h^  ~  30,05. 

What  was   the  difference  of  level  ? 
Here 

f^  ^  t'  =  148°  ;     T  —  T'  =  7°,2  ;     Latitude  21°. 

To   log     30"b5  =  1,4778445 

Add  6' for  7°,2  =  9,9990814 

1,4775259 

Sub.  locr  23,00  =  1.3740147 


Log  of     -     -     -     0,1035112  =  -  1,0149873 

Add  A  for  148°  -    -    -     -     =        4,8193975 

Add  B  for  21°    -     -    -     -     =        0,0008089 

ft.  ■ 

0843,1 3,8352537; 

whence   the    mountain    is    0843,1  feet  high. 

It  will  be  remembered  that  the  final  Equation  (404)  was  deduced 
on  the  supposition  that  the  air  is  in  equilibrio — that  is  to  say, 
when  there  is  no  wind.  The  barometer  can,  therefore,  only  be  used 
for  levelling  purposes  in  calm  weather.  Moreover,  to  insure  accu- 
racy, the  observations  at  the  two  stations  whose  difference  of  level 
is  to  be  found,  should  be  made  simultaneously,  else  the  temperature 
of  the  air  may  change  during  the  interval  between  them  ;  but  with 
a  single  instrument  this  is  impracticable,  and  we  proceed  thus,  viz. ; 
TalvC  the  barometric  column,  the  reading  of  the  attached  and  detached 
thermometers,  and  time  of  day  at  one  of  the  stations,  say  the 
lower;  then  proceed  to  the  upper  station,  and  take  the  .same 
elements  there  ;  and  at  an  equal  interval  of  time  afterward,  observe 
these  elements  at  the  lower  station  again ;  reduce  the  mercurial 
columns  at  the  lower  station  to  the  same  temperature  by  Equation 
(394),  take  a  mean  of  these  columns,  and  a  mean  of  the  tempera- 
tures   of    the    air   at   this    station,  and    use    these    means   as    a    single 


326  ELEMENTS     OF     ANALYTICAL     MECHANICS. 

set    of  observations    made    simultaneously    with    those    at    the    hijTher 
station. 

Example. — The    fl allowing    observations   were    made    to    determine 
the    height   of  a  hill    near  West  Point,  N.  Y, 

Upper  Station.  Lower  Station. 

(1)  (2) 

Detached   thermometer,  t'   —  57°  ;       t^   =  56°       and  61°. 
Attached  "  '   T'  =  57,5  ;     T  =  50,5     and  03. 

Barometric  column,  h   =  28,94 ;  h,  =  29,02  and  29"g3. 

First,    to    reduce   29,63    inches   at    63°,    to    wdiat    it   would    have 
been    at  56°, 5.     For    this   purpose,  Equation    (394)  gives 


h{l  +  T  -  T'  X  0,0001)  =  29,63  (1  -  6,5  x  0,0001)  =z  29,611 
Then 

56°  4-  61° 

t,  =  ^ .   -  -  .:.  5S°,5, 

t^  +  t'  =  5S°,5  +  57°-  -  =  115°,5, 
T—T'=:  56°,5  -  57°,5  -  =  -  1°. 

To  log  29,6155  =1,4715191 
Add  C  for  -  1°  ==  0,0000434 
1,471.5625 
Sub.  log  of  28",94  =  1,4614985 
Log  of  -  -  -  -  0,0100040  =  -  2,0027700 
Add  .4  for  115°,5  -  -  -  =  4,8048112 
Add  B  for  41°,4    ...-_::        0,0001465 


ft- 


642,28 2,8077283; 

whence  the  height   of  the  hill  is  642,28  English  feet. 

MOTION    OF    HEAVY    IN-C03*rPKESSIBLE    FLUIDS    IN    VESSELS. 

§285. — A  heavy  homogeneous  liquid  moving  in  a  vessel,  may 
be  regarded  as  an  assemblage  of  indefinitt'ly  thin  strata  ai-raiiged 
perpendicularly  to  the  direction  of  the  motion,  and  these   strata  miiy 


MECHANICS      ,^F    FLUIDS. 


327 


be  reg.'irded  as  so  many  solid  bodies,  provided  we  attribute  to 
them  the  property  of  contracting  and  expanding  in  different  direc- 
tions so  as  to  maintain  a  constant  volume  in  adapting  themselves 
to  the  varying  cross  section  of  the  vessel  in  which  they  are  moving. 

Let  ABCD  he  a  vessel  of 
which  the  axis  is  vertical,  and 
whose  horizontal  sections  vary 
only  by  insensible  degrees  ;  sup- 
pose the  fluid  divided  into  an 
indefinite  number  of  thin  level 
strata  whose  volumes  are  equal 
to  one  another.  We  may  sup- 
pose that  at  the  end  of  each 
element  of  time  any  one  stratum 
occupies  the  space  filled  by  the 
stratum  which  preceded  it  at 
the  commencement  of  this  ele- 
ment. 

Tlie  horizontal  velocities  of  the  particles  of  the  fluid  may  be 
disregarded,  and  the  vertical  velocity  of  any  one  of  them  will  be 
the  same  as  that  of  every  other  particle  in  the  same  stratum. 
The  motion  of  the  fluid  will  be  known  when  we  know  that  of  any 
one   stratum. 

§  286. — Taking  the  axis  of  z  vertical  and  positive  upwards,  we 
shall  have,  in  Equations  (400)  and  (401), 


.r  =  0 ;    r 

and,  therefore, 


0;     Z  =  —  ff  ;     li  =  0  ;     v  =  0, 


D  '  dx  ~      '     D  '  dy~ 


div 
Iz 


0; 


1      dp 

'd'^Tz 


9  + 


dxo 


dt  ' 


.,  t^^  $La. 

in  which  it  will  be  recollected  that  w  is  the  velocity  of  any  one 
particle,  and  therefore  of  the  stratum  to  which  it  belongs,  in  the 
direction  of  z. 


328         ELEMENTS     OF    ANALYTICAL     MECHANICS. 

Multiplying  the   last    equation  by  Ddz.  and  integrating,  we  have 
p^  -  D.g.z  +  D-f^-^^-dz  -{-  G.    '     .     .     (465) 

Take    the  following   notation,  viz. : — 

s    —   the  variable  area  of  the    stratum  Avhose  velocity  is  'w. 

s^  =  the  constant  area  of  any  determinate   horizontal  section  of  the 

vessel,  as   CD. 
S  =  the  area  of  the    section  of  the    vessel   by  the  upper  surface  of 

the   liquid ;  this   may  be  constant  or  variable,  according   as  the 

upper  surface  is   stationary  or  movable. 
Wj  =  velocity  of  the    stratum   passing    the    section    s^  at   CD,  at  the 

time  t. 
The    fluid    being    incompressible,    the    same    volume    must    pass 
everv^  horizontal   section  in  the   same  interval  of  time  ;  and  hence 

w  •  s^  =  w  '  s, 


and 


but 


~      s 


dw  s^     dWj  ds     dz      \ 

~dT  ~  T  ■  ~dt  '   '     dz     dt     6-2  ' 


dz  \i},Si 

dt~~       ~     s 

Substituting    this   in    the    last    term,    and    multiplying   by    dz,    we 
have 

dw      ,  dio.     dz    ^       „      „    ds 

— dz  ■=  S'  —~  • \-  S"  w/'  •  — -  ; 

dt  '      dt       s  '      '      s^  ' 

and   integrating,  regarding  z,  and   therefore  s,  as    variable, 

fi^.i,^,^Mj^Lf'h-^J!l.%     .    .    .     (406) 
J    dt  '     dt  ^    s  2       s2  ^       ' 

which,  in  Equation  (4G5),  gives 

j,  =  ^Ds-.  +  D.s,.^J--D^.^+  C..(467) 


MECHANICS     OF     FLUIDS.  329 

To  find  the  value  of  C,  let    'p  =z  P ^  ^    when    z  —  z,^  which    corre- 
sponds to   the  section   CD  of  the    liquid  ;  then  will 

which,    subtracted  from   the  ec[uatioia    above,  gives 

Also,  if  P'  denote  the  pressure  at  the  uj^pcr  surface  coi-respondiiig 
to  which  z  =  z\  we   have 

Now  z'  —  Zj  =:  h  ^=  height  of  the  lluid  surface  above  the  section 
CD',   whence,  by  substitution  and    transposition, 

P-_P,  +  ^,„_^,/iif..^;'i^_.,.|^(._|!)  =0.(470) 

The  quantity  of  fluid  flowing  through  every  section  in  the  same 
time    being    equal,  we    also   have 

—  Sdh  =  s,  .io^  .d(.      ' (4TI) 

By  means  of  this  equation,  f  may  be  eliminated  from  Equation 
(470)  ;    then    knowing  the   cjuantity  of  the  liquid,   the   size   and  figure 

y"'' d  z         P^dz 
=       /  ■ 5 

in  which  s  is   a    function  of  z. 

§287. — The  value  of    — —^   being  found  from  Equation  (470).  and 

substituted  in  Equation  (4G8),  this  latter  equation  will  give  the  value 
of  the  pressure^  at  any  point  of  the  fluid  mass  as  soon  as  w^  be- 
comes   known. 

Two  cases  may  arise.  Either  the  vessel  may  be  kept  constantly 
full  while  the  liquid  is  flowing  out  at  the  bottom,  or  it  may  be 
Buffered    to    empty  itself 

§288. — To  discuss  the  case  in  wliich  the  vessel  is  always  full,  or 
the   fluid   retains   the  same  level  by   being  supplied  at  the  top  as  fast 


830  ELEMENTS     OF     ANALYTICAL    MECHANICS. 

as   it  flows  out  at  the  bottom,  the   quantity  h  must   be  constant,  and 
Equation  (471)  will    not   be    used. 
And  making,  in  Equation  (470), 


^  =  2../ 


B  =  2o(l.  +  ^--^)- 


S  2 

r  —  iL 1 


and   solving  with  respect   to  d  t,  we   have 

-  =  ^'^. (-*) 

Now,  three  cases  may  occur.  ' 

1st.  S  may  be  less   than  s^ ,  and   C  will  be  positive. 

2d.    S  may  be  equal    to  s^ ,    hi  which    case    C  will    be  zero. 

3d.    S  may  be  greater  than  5^ ,  when   C  will  be  negative,  and  this 

is   usually  the  case  in  pi-actice. 
In  the  first  case,  when  0  is  positive,  we  have,  by  integrating  Equa- 
tion (472),  and  supposing   t  =  0,  when  w^  =  0, 

^  -\         [G  ,_„, 

i  =:  — : ■  •  tan     w.  \  /  —  : (473) 

■x/jJC  '  y  B  ' 


whence, 


tc^  =^-.tan  -^^—p.^. (474) 

from  which  we  see  that  the  velocity  of  egress  increases  rapidly  with 
the   time ;    it   becomes    infinite  when 


or 


i=      ^'A- (475) 

2^  BO 

When   (7=0,  then  will   the   integration  of  Equation  (472)  give 

t  =  ^^-^^n •     •     •     (4W) 


MECHANICS    OF     FLUIDS.  331 

or  replacing  A  and  B  by  their  values,  and  finding  the  value  of  w^ , 

-'  =  P^t/— -^5 (477) 

whence,  the  velocity  varies  directly  as  the  time,  as  it  should,  since 
the  whole  fluid  mass  would  fall  lilvc  a  solid  body  under  the  action 
of  its  own  weight. 

When   C  is   ne2;ative,  the    inteo;ration  slves 


A 


whence. 


e     •"        -1  IB 


c     -^        +1 
in  which  e  is  the  base  of  the  Naj)erian  system  of  logarithms  =  2,718282. 
If  the  section    S   exceeds   s,  considei'ably,  the  exponent   of  e  will 
soon    become  very  great,  and  unity  may  be  neglected  in  comparison 
with    the    corresponding    power  of  e  ;  whence, 


Ic  ^'-U 


'.,(,.,^1^) 


B  /       -^    \       '  Uq 

'''  =  V6^  =  V V^ ■'•    •    •    (^'^) 

^  ^      ^  1  _  ^ 

that  is   to    say,  the  velocity  will    soon  become  constant. 

If  the  pressure  at  the  upper  surface  be  equal  to  that  at  the  place 
of  egress,  which  would  be  sensibly  the  case  in  the  atmosphere, 
P'  —  P,  =  0,  and 


2gh 


1  _^; (480) 

and  if  the  opening  below  become  a  mere  orifice,  the  fraction 

and 

w,  =  V^TA; (481) 


332  ELEMENTS     OF    ANALYTICAL    MECHANICS. 

that  is  to  say,  the  velocity  wilh  wliieh  a  hofivy  licjuid  will  is^:ue 
fVom  a  small  orifice  in  the  bottom  of  a  vessel,  -svhen  suhje'ctoi]  to 
the  pressure  of  the  superincumbent  mass,  is  equal  to  that  aeijuii-ed 
by  a  heavy  body  in  falling  through  a  'height  equal  to  the  depth  v>f 
the  orifice  below  the  upper  suifacc  of  the  liquid.  The  veloi'ities 
given  by  Ecpations  (479),  (480),  (481),  are  independent  of  the 
figure  of  the  vessel. 

If  the  velocity  w^  be  multiplied  by  the  area  s^  of  the  orifice,  the 
product  will  be  the  volume  of  fluid  discharged  in  a  unit  of  time. 
This  is  called  the  expense.  The  expense  multiplied  by  the  time  of 
flow  will   give  the   whole    volume    discharged. 

§289. — The   velocity   u\   being   constant  in   the   case    referred  to  in 
Ec[uation  (479),  we   shall  have 

—  -  =  0, 
lit 

and  Equation  (4G8)  becomes 

V^P,-  Dg  (.  -  .■ )  _  i>  .  ^-  .  (^  _  l)  , 
or,   substituting  the  value   of   iv^ ,  given    by  Equation  (470), 


2 

^  -  1 


C  s 


p  =  P^-  Dg  {z  -  .J  +  {^Bgh  4-  P'  -  B)  -^ ;    .     .     (482) 

'  '         1 

\\heuce,  it  appears,  that  when  the  flow  has  become  uniform,  the  pres 
sure  upon  any  stratum  is  wholly  independent  of  the  figure  of  the 
ve.ssel,  and  depends  only  upon  the  area  s  of  the   stratum,  its  distance 


from   the  upper   surface  of   the  fluid,  and   upon  the  ratio -^ 


^2 


ifO 


§290.— ]f  the  vessel  be  not  replenished,  but  be  allowed  to  empty 
itself,  h  will  be  variable,  as  will  also  S  except  in  the  particulaJ 
cases  of  the  prism  and  cylinder. 

Making 

'.  =  V^Vh; (483) 


w. 


MECHANICS     OF     FLUIDS.  333 

in   wliuih  H  denotes   the  height  due  to   the  velocity  of  discharge :  we 

have 

3-dH  ,       ^ 

dio,  =  ^^7==; (484) 


and,  Equation  (471), 

S  •  d  h  ,  ,^ 

d^= 7=^; 485) 


1           rS-dh  ,_„, 

t  ^  C y^  '   /  — -= (486) 


and  by  integration, 


To  effect  the  integration,  S  and  H  must  be  found  in  terms  of  h. 
The  rehition  between  S  and  h  will  be  given  by  the  figure  of  the 
vessel.  Then  to  find  the  relation  between  H  and  A,  eliminate  xv^ , 
d  w^ ,  and  d  t  from  Equation  (470),  by  the  values  above,  and  we  have 


o  2  ph 

%■     f 

S       Jo 


i',2  dff  r^  d 

-^  ^  k)  .  cUi  + 

0 

or,  dividing  by 

s,2       pT^  dz 


S 


-M-O  ^-(i-l!) 


^  J^f~       '        "^  77        ,         ITT  "    ■     \"      ~     S^ 


dh  +  dH ^-f H'dh=0-  (487) 


,^2  r  ^ii  ,^2  /^'^  ^!f 

'Jus  'Jos 

and  making  "^  '' 


5.(1-^)  s.(^^+i)     .a     . 


S  •    1/  o       s 


^^  dz        '      ^  ^    P^'  d  z  '  '    -'A 

,  /'''    ^'  <2^^i  +  f?^+  RHdh  =  0.       ...        ,(488) 

Mult' plying  by  e         , 

/if  rf  A  flldh  /r  (1  h 

dh  •  Q     e  -\-  dll  •  a  \-  ji  .  e  X  JR  d  h  =  0  ; 


534  ELEMENTS     OF    ANALYTICAL    MECHANICS. 


or 


and  integrating 


jRdh  .        fRdh 

dh-  Q  •  e  +  d  (He         j  =  0; 


fdh  '  Q 


fRdh  fRdh 

e         +  He         =  C;  .     ,     .     .     (489) 


whence. 


—fRa/i  fRdh^ 

H  =  e        •    {C  -   fdh-Q  .  e        \ 


(490) 


The  constant  must  result  from  the  condition,  that  when  H  =  0, 
h  must  be  h, ,  the  initial  height  of  the  fluid  in  the  vessel. 

Thus  H  becomes  known  in  terms  of  A,  and  its  value  substituted 
in  Equation  (486)  will  make  known  the  time  required  for  the  fluid 
to  reach  any  altitude  h.  The  constant  in  Equation  (486)  must  be 
determined,  so  that  when  ^  =  0,  h  =z  h^. 

§291. — The  mode  of  solution  here  indicated  is  direct  and  general; 
but  analysis,  in  its  application  to  the  motion  of  fluids,  often  pre- 
sents itself  under  forms  which  require  us,  in  particular  cases,  to 
adapt  the  mode  of  solution  to  the  peculiarities  which  belong  to  them. 
Take,  for  example,  the  case  of  a  right  cylinder  or  prism.  Here  S 
will  be  constant,  and  equal  to  s. 

^h     dz  _    h 
1'   ~  J' 


/: 


Moreover,  let  us  suppose  F'  —  F^  =  0,  which  would  be  sensibly 
true  were  the  fluid  to  flow  into  the  atmosphere   that  rests  upon  its 

S 
upper  surface.     Also,  for    the    sake    of    abbreviation,  make  —  =:  k, 

then  will 

/;2  -   1         1  -  F     . 


^   =: 


aud 


h 
dh 


jRdh  =  (1  -  ^^fj  =  (1  -  '^^•^ )  log  h. 


MECHANICS    OF    FLUIDS.  3.55 

and  Eq.  (490)  becomes 

-(l-i2)logA  (l-;c:)lc?ft 

H  =  e  •  [  C  -  JkMh  •  e  J 

Multiplying  the  last  t-n-m  by 

we  may  write 

-(l-^a)log  <,     r  p  ^  (1_42;  log  A 


_(i_ft2)iog^    p  p  (1-42;  log  A -r 

-(i-S;2)iog  ft    r  ^,2  ;i-fc^)iogA-i 


when  //=  0,  then  will  h  =  h,   and 


JA  U— 'c";  In?  A, 


but, 


which  substituted  above,  gives,  after  rediict'on,  ^\h,  ■ 

and  therefore, 

^=2^fL(t)         -U^A;^"2L'-(a:)       J    •     •     •     (^92) 

which    substituted   in   Equation  (48G),  gives 

in   which    the    only  variable  is  h. 

§  292.— The  particular  case  in  which  F  =  2,  gives  to  this  value 
for  t  the  form  of  indetermination.  When  this  occurs,  we  must  have 
recourse  to  the  form  assumed  by  Equation  (488),  which,  mider  this 
supposition,  becomes 

2hdh  +  hdH  —  Hdh  z=  0  : 


336  ELEMENTS     OF     ANALYTICAL    MECHANICS. 


multiplying   by  h    ", 

IhT^dh  +  /r\  dH  -H.h'^dh  =z  0, 
dh  W 

2  log  A  +  —  =  C; 

and  because  I£  =  0  when  A  =  h^ , 

2  log  A,  =  C;  /;   ^12  li<^  )t  =  (^ 

whence. 


H  =2h-\og  4^, 


and  this,  in  Equation  (486),  gives 


//'  -    2    /.  -{/A^i 


ju     /.  c?A 

t  —  C 


V^ 


y 


^/2A.logA 


^^     S    el  A 

Makmg  — ^  =  — -5  this  becomes  '^^ 

^    A  a;2 


0     J     L        1 


The   value  of   C  is   determined   by   making  .r  =  1  when  <  ::=  0. 


§  293. — If  the  orifice  be  very  small  in  comparison  with  a  cross 
section  of  the  prismatic  or  cylindrical  vessel,  then  will  H  =  A,  and 
Equation  (486)  gives 

Making   i  =  0   M^hen  A  =  A^ ,  we   have 

t=^^•{^f\-^^^ (494) 

and  for  the  time  required  for  the  vessel  to  empty  itself,  A  =  0,  and 

^-— V^ (495) 


MECHANICS     OF     FLUIDS.  337 

Now,  with    the    same   relation  of  the    orifice    to    the   cross   section 
of  the  cylindrical  vessel,  we   have,  Equation  (481), 

and   for   the    volume    of    fluid    discharged   in   the    time    t    when    the 
vessel    is    kept  full, 

w^  .  s^ .  t  ::=  s^ .  t .  y2ffh, 

and    if  this   be    equal    to   the    contents    of  the   vessel, 


whence, 

_  ;?       /I7 

6^        *    ~  y 
Tliat   is.  Equation  (495),  the   time  required  for    a  prismatic  or  cylin- 
drical   vessel    to    discharge    itself    through    a    small    orifice     at   the 
bottom    is    double    that    required    to    discharge   an    equal    volume,  if 
the  vessel  were   kept  full. 

§  294. — The     orifice    being    still    small,    we     obtain,    from    Equa- 
tion (485),  v: 

whence  it  appears  that,  for  a  cylindrical  or  prismatic  vessel,  the 
motion  of  the  upper  surface  of  the  fluid  is  uniformly  retarded.  It 
will  be  easy  to  cause  S  so  to  vary,  in  other  words,  to  give  the 
vessel  such  figure  as  to  cause  the  motion  of  the  upper  surface  to 
follow  any  law.  If,  for  example,  it  were  required  to  give  such  figure 
as  to  cause  the  motion  of  the  upper  surflice  to  be  imiform,  then 
would  the  first  member  of  the  above  equation  be  constant ;  and, 
denoting    the  rate  of  motion   by  a,  we    should   have 


whence, 

2  .  2  V  h 


S^ 


,2 


but  supposing    the    horizontal    sections    circular. 


33S  ELEMENTS     OF     ANALYTICAL     MECHANICS, 

and,  therefore, 


=VW'/^-' 


whence  the  radii  of  the  sections  must  vaiy  as  the  fourth  root  of 
their  distances  from  the  bottom.  These  considerations  apply  to  the 
construction  of  Clepstjdras  or    Water   Clocks. 


k^ 


MOTION   OF   ELASTIC   FLUIDS    IN    VESSELS. 

§295. — As  in  the  case  of  incompressible,  so  also  in  that  of 
elastic  fluids,  it  is  assumed  that  in  their  movement  through  vessels, 
they  arrange  themselves  into  parallel  sti'ata  at  right  angles  to  the 
direction  of  the  motion.  The  quantity  of  matter  in  each  stratum 
is  supposed  to  remain  the  same,  while  its  density,  which  is  always 
uniform  throughout,  may  vary  from  one  position  of  the  stratum  to 
another ;    hence,  the    volume    of  each    stratum  may  vary. 

All  lateral  velocity  among  the  particles  will  be  supposed  zero  ; 
and '  as  the  weight  of  the  elements  of  elastic  fluids  is  insignificant 
in  comparison  to  their  elasticity,  the  former  will  be  disregarded. 
The  motion  Avill,  therefore,  be  due  only  to  the  elastic,  force  arising 
from  some  force  of  compression  ;  and  as  the  fluid  wUT  be  supposed 
to  communicate  freely  with  the  air,  or  with  a  vessel  partly  filled 
with  some  other  elastic  fluid,  this  force  within  may  be  greater  or  less 
than   it   is   on   the    exterior  of  the    vessel. 

§290. — Assuming    the    axis    of    the    vessel    horizontal,    take    that 
line  as  the  axis  of  x. 
Then,  by  the  supposi- 
tion above,  will 

X  =0; 

Z  =  0  ; 
v  =  0; 
w  —  0: 


Z 


A 


Al 


D 


X 


Ji 


2i' 


MECHANICS     OF    FLUIDS.  339 

and  Equations  (400)  give 

1  .4^  J_(4!i)_4^.^.   ....     (496) 
D      dx  \(lt  /         dx  ^       ' 

Moreover,  if  we  suppose  the  motion  to  have  been  established 
and  become  permanent,  the  velocity  of  a  stratum  as  it  passes  any- 
particular  cross  section  of  -the  vessel  will  always  be  constant,  and 
the  quantity  of  fluid  which  flows  through  every  cross  section  will 
be  the  same.  Hence  the  partial  differential  of  u  in  regard  to  the 
time,  that  is,  supposing  x,  y,  2,  to  be  constant,  must  be  zero,  and 
tne   above   equation  reduces   to 

dp  =  —  D  .u  .  du. 
From    Mariotte's   law,  Equation  (389), 
p^P.D, 
and   by    division. 


and   by  integration, 


dp  1 

—   =  —  —  '  ltd  n, 

p  F 


logl^=  ^-Wp"'^-    •.'•••     (407) 


To  determine  the  constant,  let  p,  be  the  pressure  at  the  opening 
CI),  that  is,  the  pressure  of  the  atmosphere,  and  denote  by  u^  the 
velocity  of  the    fluid   at   this   point,  then  will 


log;?,  =  C  -—• 


u, 


and   by    subtraction. 


log^  =  ^.(V-"^)- (498) 

Denote  by  s  the  area  of  any  section  of  the  vessel  A'  B\  at  which 
the  pressure  is  p  and  velocity  u,  by  D  the  density  of  the  fluid  at 
this  section,  and  by  D,  that  at  the  section  CD  equal  to  s^ .  Then, 
since  the  quantities  of  fluid  flowing  through  these  sections  in  a  unit 
of  time   must  be   equal,  we   have 

D  .  s  .  u  =z  D, .  s,  .u,\ 


34:0         ELEMENTS     OF    ANALYTICAL    MECHANICS, 
but,  §244, 


whence. 


which,  in  Equation  (498),  gives 


loff 


p,        2P 


p .  s 


1 


(!0} 


(499) 


If  2^'  denote   the    pressure  exerted    by  the    piston  A  B,  and  *S'   de- 
note  its   area,  we   have 


^  p^  -  2pL  \p'  S/  J' 


whence. 


2P.loiT 


P< 


\p'  sy 


(500) 


(501) 


This    is    the   velocity    with    which    the    fluid    will     issue    into .  the 
atmosphere  or  other  fluid  whose  pressure  on  the  unit  of  surface   is  jy, . 

§  297. — The  volume  discharged  in  a  unit  of   time  is 


u  s    —  y 


2  P  •  log  ^ 


_  (PiIj\ 


while    under   the    pressure  2^/  '■>    fi"d    under    a   pressure    equal    to    that 
on    the    unit    of   surface    of  the    piston, 
or  top  of  a  gasometer,  and  which  would 
be  indicated  by   a  gauge,  since  the  vol- 
umes are  inversely  as  the  pressures. 


',.  =  ^-., 


M  p  •  lo" 


p, 


\p'  s) 


(502) 


MECHANICS    OF    FLUIDS.  341 

i^SyS. — Dividing  Eqiuition   (499)  hy   Ec|uation   (500),    .ve  luive 


-^  = ^'^^  :.....     (503) 

which  will  give   the  pressure  'p  at  any   section  of    the  vessel. 

§299. — If  the  opening   CD  is  very  small  in  reference  to  ^4  ^,  the 
velocity  it^  ■will   Become,  Equation    (501), 


,,=^^2P.\og^- (.501) 


Pi 

and    the  volume   of  fluid  discharged  in  a  unit  of  time  and  of  a  den- 
sity equal  to  that  pressing  upon  the  gauge, 


and  Equation  (503)  becomes 


2P  -log^; (50.5) 

Pi 


&■ 


S  300. — A  stream  flowinor  through  an  orifice  is  called  a  vein.  In 
estimating  the  quantity  of  fluid  discharged,  it  is  supposed  that  there 
are  neither  -within  nor  without  the  vessel  any  causes  to  obstruct  the 
free  and  continuous  flow ;  that  the  fluid  has  no  viscosity,  and  does 
not  adhere  to  the  sides  of  the  vessel  and  oi'ifice  ;  that  the  particles 
of  the  fluid  reach  the  upper  surface  Avith  a  common  velocity,  and  also 
leave  the  orifice  with  ecjual  and  parallel  velocities.  None  of  these 
conditions  are  fulfilled  in  practice,  and  the  theoretical  discharge  must, 
therefore,  differ  from  the  actual.  Experience  teaches  that  the  former 
always  exceeds  the  latter.  If  we  take  water,  for  example,  which  is 
far  the  most  important  of  the  liquids  in  a  practical  point  of  view, 
we  shall  find  it  to  a  certain  degree  viscous,  and  always  exhibiting  a 
tendency  to  adhere  to  ununctuous  surfaces  Mith  which  it  may  be 
brought    in    contact.       When    water    flows     through    an     opening,    the 


34-2  ELEMENTS     OF     AXALYTICAL     MECHANICS. 

adhesion  of  its  particles  to  the  surflxce  will  check  their  molii)n,  anJ 
the  viscosity  of  the  fluid  will  transmit  this  effect  towards  the  interior 
of  the  vein;  the  velocity  will,  therefore,  be  greatest  a  the  axis  of 
the  latter,  and  least  on  and  near  its  surface ;  the  inner  particles  thus 
flowing  away  from  those  without,  the  vein  will  increase  in  length  and 
diminish  in  thickness,  till,  at  a  certain  distance  from  the  orifice,  the 
velocity  becomes  the  same  throughout  the  same  cross-section,  which 
usually  takes  place  at  a  short  distance  from  the  aperture.  This 
eff'ect  will  be  increased  by  the  crowding  of  the  particles,  arising  from 
the  convergence  of  the  j)aths  along  which  they  approach  the  aper- 
ture, every  particle,  which  enters  near  the  edge,  tending  to  pass 
obliquely  across  to  the  opposite  side.  This  diminution  of  the  fluid 
vein  is  called  tlic  veined  contraction.  The  C|uantity  of  fluid  discharged 
must  depend  upon  the  degree  of  veinal  contraction,  and  the  velocity 
of  the  particles  at  the  section  of  greatest  diminution ;  and  any  cause 
that  will  diminish  the  viscosity  and  cohesion,  and  draw  the  particles 
in  the  direction  of  the  axis  of  the  vein  as  they  enter  the  aperture, 
will  increase  the  discharge. 

Experience  shows  that  the  greatest  contraction  takes  place  at  a 
distance  from  the  vessel  varying  from  a  half  to  once  the  greatest 
dimension  of  the  aperture,  and  that  the  amount  of  contraction  de- 
pends somewhat  upon  the  shape  of  the  vessel  about  the  oViiice 
and.  the  head  of  fluid.  It  is  further  found  by  experiment,  that  if  a 
tube  of  the  same  shape  and  size  as  the  vein,  from  the  side  of  the 
vessel  to  the  place  of  greatest  contraction,  be  inserted  into  the 
aperture,  the  actual  discharge  of  fluid  may  be  accurately  computed 
by  Equation  (478),  provided  the  smaller  base  of  the  tube  be  sub- 
stituted for  the  area  of  the  aperture ;  and  that,  generally,  without 
the  use  of  the  tube,  the  actual  may  be  deduced  from  the  theoretical 
discharge,  as  given  by  that  equation,  by  simply  multiplying  tlie 
theoretical  discharge  into  a  co-efficient  whose  numerical  value  depends 
upon  the  size  of  the  aperture  and  head  of  the  fluid.  Moreover, 
all  other  circumstances  being  the  same,  it  is  ascertained  that  this 
co-efficient  remains  constant,  whether  the  aperture  be  circular,  square, 
or  oblong,  which  embrace  all  cases  of  practice,  provided  that  mi 
comparing  rectangular  with  circular  orifices,  we  compare  the  smallest 


MEOHANICS     OF    FLUIDS. 


348 


dimension  of  the  formei'  with  the  diameter  of  the  latter.     The  value 

of  this    co-efficient   depends,   therefore,  when    other   circumstances   are 

the    same,    upon    the    smallest    dimension    of    the   rectangular    orifice, 

and    upon  the  diameter  of  the   circle,  in  the  case  of  circular  orifices. 

But    should    other   circumstances,,  such    as    the   head   of  fluid,  and  the 

place  of  the   orifice,  in    respect   to  the  sides 

and    bottom    of  the    vessel,    vary,   then    will 

the    co-efficient   also   vary.      When    the   flow 

takes   place    through  thin  plates,  or   through 

orifices  whose   lips   are    bevelled    externally, 

the  co-efhcient  corresponding  to  given  heads 

and    orifices,    may    be    found     in    Table    V, 

jDrovided   the    orifices    be    remote    from    the 

lateral    faces    of    the   vessel.      This    table    is 

deduced    from    the    experiments    of   Captain 

Lesbros,  of  the  French  engineers,  and  agrees 

with    the   previous   experiments    of  Bossut,  Michelotti,  and  others. 

As  the  orifice  aj)proaches  one  of  the 
lateral  faces  of  the  reservoir,  the  contrac- 
tion on  that  side  becomes  less  and  less, 
and  will  ultimately  become  nothing,  and  the 
coefficient  will  be  greater  than  those  of  the 
table.  If  the  orifice  be  near  two  of  these 
faces,  the  contraction  becomes  nothing  on 
two  sides,  and  the  co-efficient  will  be  still 
greater. 

Under  these  circumstances,  we  have  the 
following  rules  : — Denote  by  C  the  tabular, 
and  by  C  the  true  co-efficient  corresponding 
to  a  given  aperture  and  head ;  then,  if  the 
contraction   be    nothing  on    one    side,  will 

C  =  1,03  C; 
if  nothing   on    two    sides, 

C  =  1,06  0; 
if  nothing    on    three    sides, 

C  ^  1  12  C; 


mi 


344  ELEMENTS     OF     ANALYTICAL     TJECHANICS. 

and  it  must  be  borne  in  mind,  that  these  results  and  those  of  the 
table  are  applicable  only  when  the  fluid  issues  through  holes  in 
thin  plates,  or  through  apertures  so  bevelled  externally  that  the 
particles  may  not  be  drawn  aside  by  molecular  action  along  their 
tubular   contour. 

§301. — When  the  dischai'ge  is  through  thick  jjlutes  wilhotit  bevel, 
or  through  cylindrical  tubes  whose  lengths  are  from  two  to  three 
times  the  smaller  dimension  of  the  orifice,  the  expense  is  increased, 
the  mean  coefficient,  in  such  cases,  augmenting,  according  to  experi- 
ment, to  about  0,815  for  orifices  of  which  the  smaller  dimension 
varies  from  0,3o  to  0,66  of  a  foot,  under  heads  which  give  a  coeffi- 
cient 0,619  in  the  case  of  thin  plates.  The  cause  of  this  increase  is 
obvious.  It  is  within  the  observation  of  every  one,  that  water  will 
wet  most  surfaces  not  highly  polished  or  covered  with  an  imctuouo 
coating — in  other  words,  that  there  exists  between  the  particles  of 
the  fluid  and  those  of  solids  an  aflinity  which  will  cause  the  former 
to  spread  themselves  over  the  latter  and  adhere  with  considerable 
pertinacity.  This  affinity  becoming  eflective  between  the  inner  sur- 
face of  the  tube  and  those  particles  of  the  fluid  which  enter  the 
orifice  near  its  edge,  the  latter  will  not  only  be  drawn  aside  from 
their  converging  directions,  but  will  take  with  them,  by  the  force  of 
viscosity,  the  other  particles,  with  which  they  are  in  sensible  contact. 
The  fluid  filaments  leading  through  the  tube  will,  therefore,  be  more 
nearly  parallel  than  in  the  case  of  orifices  through  thin  plates,  the  con- 
traction of  the  vein  will  be  less,  and  the  discharge  consequently 
greater. 


PART    III. 


MECHANICS  OF  MOLECULES. 


§  302. — The  move  general  circumstances  attending-  tlie  action  of 
forces  upon  bodies  of  sensible  magnitudes  have  been  discussed.  They 
constitute  the  subjects  of  Mechanics  of  Solids  and  of  Fluids.  Tliose 
which  result  from  the  action  of  forces  upon  the  elements  of  both  soliils 
and  fluids  remain  to  be  considered.  They  form  the  subject  of  Me- 
cJi'iiiics  of  Molecules ;  which  con:iprchends  the  whole  theory  oi  Electrics, 
T/'ermotics,  Acoustics,  and    Optics. 

It  has  been  seen,  that  all  bodies  are  built  up  of  elementary  mole- 
cules in  sensible,  though  not  in  actual,  contact;  that  the  relative  places 
of  equilibrium  of  these  molecules  are  determined  by  the  molecular  forces, 
and  that  the  intensities  of  these  forces  are  some  function  of  the  dis- 
tance between  the  acting  molecules.  A  displacement  of  a  single  mole- 
cule from  its  position  of  relative  rest,  will  break  up  the  equilibrium  of 
the  surrounding  forces,  and  give  rise  to  a  general  and  progressive  dis- 
turbance throughout  the  body.  It  is  proposed  to  investigate  the  nature 
of  this  disturbance,  the  circumstances  of  its  progress,  and  the  conduct 
of  the  molecules  as  they  become  involved  in  it. 

PERIODICITT    OF    MOLECULAR    CONDITIOlSr. 

§  303. — Molecular  motions  cannot,  like  the  initial  disturbances  which 
produce  them,  be  arbitrary ;  but  must  fulfil  certain  conditions  imposed 
by  the    physical    connections  which   unite  the    molecules  into  a  system. 


346  ELEMENTS    OF    ANALYTICAL    MECHAXICS. 

These  motions  are,  so  to  speak,  constrained  by  this  connection.  Let 
the  conditions  of  constraint  be  expressed,  as  in  §  213,  Mecli.  of 
Solids,  by 

Z  =  0  ;     Z'  =  0  ;     Z"  =  0  ;    etc (506) 

Z,  Z',  Z",  (fcc,  being  functions  of  the  co-ordinates  of  the  molecules, 
Denote  by 

X,  r,  Z ;  .Y',  Y\  Z' ;    kc, 

the  accelerations  impressed  upon  tlie  molecules  ^vhose  masses  are  m,  m\ 
<fec.,  in  the  directions  of  the  axes.  Equation  (313)  will  obtain  for  each 
molecule.  There  will  be  as  many  equations  as  molecules,  and  by  addi- 
tion, we  find,  by  inverting  the  terms, 

^■'^[(^ - ^'^  + &- ^'^' +  (!r§ -')'']  =  '' ^'''^ 

There  will  be  three  co-ordinates  for  each  molecule.  Denote  the 
number  of  molecules  by  «';  tlie  number  of  Equations  (506)  of  condition 
by  ni ;  then  will  3i  —  ?»  :=  n,  be  the  number  of  co-ordinates  which, 
being  given,  will  reduce  the  number  of  unknown  co-ordinates  to  the 
number  of  equations.  These  unknown  co-ordinates  may,  hence,  be  found 
in  functions  of  the  known,  and  the  places  of  the  molecules  at  any  in- 
stant determined. 

Denote  the  m  co-ordinates  by  x  ij  s,  x' y' z\  «fcc.,  and  the  n  co-ordi- 
nates by  oi  (3  y,  a'/3'y',  (fcc:    then  we  may  write, 

X  =  9^  (a  [3  y  a\  (tc.)  =  ]).  ; 
y  =  9y  (a  ^  y  a',  ^-c.)  =  -p,  ; 
2  =  (p^  (a  /3  y  a',  <fcc.)  =  p^  ; 
x'=  9,-  (a  /3  y  a',  &c.)  =  }\. ; 
etc.  =  etc.  =  <fcc. ; 

also, 

X=i/;,(a/3ya',&c.)  =  P,; 

F=V.(^-'-i3r<^'c.):=P,; 

Z  =  V),  (a,3y«',  A:.-.)  =  P, ; 

X'^^\,^.{rj.^yo.\k<l)  =  P^r^ 

tfcc.  =  (fcc.  ^  (fcc. ; 

in  which  9^,  9^,,  9^,  <fcc.,  i/>„  '\\:^  1/.,  «fcc.,  denote   any  functions  of  the   co- 


MECHANICS    OF    MOLECULES. 


m 


ordinates  a  /3  y,  a'  (tc,  which   result   from   the   cocditions  of  Equations 
(50C)  and  the  process  of  elimination. 
xVt  any  time  t,  suppose 

a  ft  and  y  to  become  a  +  ^,  ./J  +  ?/,    y  +  ^; 

u.'  P'  and  y'  to  become  a'+  ^',  /3'-f7/',  y'+  ^' ; 


and  suppose  the  increments  f  ??  ^,  ^'  tj'  tf',  itc,  to  continue  so  small 
during  the  entire  motion  as  to  justify  the  omission  of  all  terms  into 
■which  their  second  powers  and  products  enter ;    then  will 

.  =  ;,,  +  ^.?  +  'if.,,  +  ^.f+i-c,icJ 
da.  a  l3  ay 


y  =  Py  + 


d  u. 


d  p,. 


+  -^  .  ^^  +  &<^;  (ire. 
dy 


z  =  p,+ 


^.S-  +  '44.,  +  n^.f+i-o.,  A-c. 


d  a 


dr,. 


dp^ 
dy 


\  .    .  (508) 


dl3 


J 

,.      „       dP,     ^  .  dP,  dP,    ^ 

o  a  d  ji  d  y 

^       ^        dP^     „       dP„  f^A     >.  ,     f 

Z.P.  +  ^-^..^+^^.,4-^.^+<^^c.,.-c., 
d  a  d  i3  d  y 


.Y'  =  P,.+ 

From  Equations  (508)  we  have 

d^:c  =  if^  .  d-^  ^  +  '^  .  d^rj  +  ^-/^  •  d-^;  +  i-c,  i-c, 
da  d  (3  dy 

a  a  rf /3  ay 


dp^ 
d  a 


dp. 

'dj 


dy 


f^'  ^  +  ^  •  ^'  ^  +  -¥r-  •  ^'  ^  +  «^'<^-»  <^^c., 


d^x'=         *tc, 


&c> 


i'C. 


(509) 


(510) 


54S 


ELEMENTS     OF    ANALYTICAL     MECHANICS. 


also  IVoin  the  saim 


6x  = 

6z 


d  p^ 
d  a 

d  p„ 
d  a. 


d  i-i 
dp., 

"^  Pz 


S'>1  + 


d  px 
d  y 


dy 


(^  Pz 

da  dp 


6v  + 


d  y 


6^-\-  &c., 

6  (^  +    etc, 


(511) 


The  Equation  (507)  contains  three  times  as  Uiauy  terms  as  there 
are  molecules,  each  term  consisting  of  a  variation  \vith  its  coefficient. 
Eliminate  from  tliis  equation  A",  Y^  Z,  A'',  etc.,  cr' .r,  (/-y,  d'Z,  d'- x\  etc., 
and  J.r,  dy,  6 z,  6x\  Arc,  by  means  of  Equations  (509),  (510),  and  (511); 
collect  the  coefficients  of  6  ^,  6  ?/,  6  ^,  6  f ',  tkc. ;  the  number  of  terms 
will  reduce  to  n,  this  being  the  number  of  the  co-ordinates  a,  /3,  y,  a', 
ike.  These  variations  are  independent  of  one  another,  since  the  co- 
ordinates a,  /3,  y,  a',  &c.,  are  so.  The  coefficients  of  these  variations 
must,  therefore,  be  separately  equal  to  zero.  Performing  the  opei-ation, 
omitting  all  the  terms  containing  products  and  powei's  of  t,  1],  i^,  s', 
ic,  higher  than  the  first,  there  will  result  n  equations  of  the  form. 


^i>.'^.+^E 


•'   ^I^'+SjP. '-1^  +  26". J+-2//..,+i:A'.;+J=0;  (r.U) 


in  which  D,  E,  F,  6-',  H,  &c.,^are  functions  of  the  ditterential  co-efficients 
in  Equations  (509),  (510),  and  (511);  and  A  consists  of  a  series  of  terms 
each  composed  of  two  factors,  one  of  which  is  either  P^,  P,j^  P,,  or 
some  other  P  with  subscript  co-ordinate  accented. 

If  a  /3  y,  «.',  ttc,  give  the  places  of  rest  of  the  molecules,  then  will 
P^^:^  Py  ^:z  P^^=  &c.  =  0,  and  Equations  (512)  become 


These  equations  are  satisfied  by  making 


R.N 


-^  n  —  r\ 


^.sin(<.  Vp-  r), 
7^  .  N^  .  sin  (/ .  ^/p  -  r),    ^ 
A' .  iV. .  sin  {i  .^~p—  ?•), 


(514) 


MECHANICS     OF     MOLECULES.  349 

ill  which  R  and  r  are  arbitrarv  constants,  and  p,  iV,,  X  ,  N ,  ito.,  are 
constants   to   be   determined.      For,  after    two   differentiations,  regardinfj' 

f,  T^j  Cj  &c.,  and  t  variable,  we  have 

^  =  -«.i\'-,.sm(iv'p->-)p, 

^  :=  -i^.iY^.  sin  (<-v/p-r)p, 

&c.,  &c. 

which,  substituted  in  Equations  (513),  give,  after  dividing  out  the  com- 
mon factor  R  .  sin  [t  y  p  —  r), 

{1:D  .N^-\-^E.N  -^^ FN )p - 2  GN^  -  i //.^^  -  ^KX  =  0.  (515) 

^  k  ^  S  I  »;  \  ^ 

Now,  there  being  ?»  of  these  equations ;  it  —  1  of  them  will  give  the 
values  of  iY ,  iY,  N'  etc.,  in  terms  of  iY^;  and  these  being  substituted 
in  the  Jt""  equation,  must,  from  the  form  of  the  equations,  give  a  result- 
ing equation  having  N ,  as  a  common  factor  and  of  the  n"'  degree  in  p. 
The  common  factor  N^  will  divide  out.  The  quantity  p  will  have  ii 
values.  The  values  of  N .  N^.  iY,,  ire,  will  be  rational  fractions  of 
the  (/J  — 1)'*"  degree  in  p,  having  a  common  denominator,  and  each  multiplied 
by  N ,  which  is,  as  yet,  arbitrary.  Make  N  equal  to  the  common  de- 
nominator, and  iV ,  N  ^  iY,,  &c.,  will  be  expressed  in  symmetrical  func- 
tions of  p,  of  the  (?i  —  1)"'  degree.  Each  of  the  quantities  iY,  N ,  N  ^ 
&c.,   will    have    as    many    values    as    p ;    and    each    of   the    increments 

g,  ?;,  1^,  ^',  <fec.,  will  also  have  n  values,  each  set  of  which  will  satisfy 
Equations  (513). 

But  Equations  (513)  are  linear;  not  only,  therefore,  will  each  of 
the  values  of  ^,  ?/,  ^,  ^',  ttc,  satisfy  them,  but  their  respective  sums  sub- 
stituted for  f,  7/,  1^,  ^\  &c.,  will  also  satisfy  them. 

Denoting  the  roots  of  the  ?i"'  equation  in  p  by  p,  p,,  p^,  <tc.,  and 
using  the  subscript  figures  to  designate  the  corresponding  values  for 
the  other  letters,  the  genei'al  solution  of  Equations  (513)  will  be 


350 


ELEMENTS     OF     ANALYTICAL     MECHANICS. 


ri=E.A'  .  sin (fl-s/fT-r) 4-/1 1.;\^   .sin(^v/pi-';'i)+-ff2--Al   .»hi{t.s/ p^-rnJ-j-Ac. 
^  ^i  '2 

l=E.]V  . sm {t.-yj- ?■)-]- J2 1. N'  .sm{f.y/'p-?\)i-Ii2.]V   .sm{t.^p'.-ri)-\-&c. 


\  (&16) 


E,  Jit,  tfec,  and  r,  r,,  (fcc,  are  arbitrary  constants,  in  these  complete  in- 
tegrals. They  must  be  found  in  terms  of  the  initial  values  of  f,  7j,  ^, 
ifec,  and  their  differential  coefficients.  They  are  small,  because  the  ori- 
ginal disturbance  is  supposed  small. 

§  304. — If  all  the  quantities  B,  J?,,  i^^,  &c.,  except  the  first,  vanish, 
Equations  (516)  lose  all  their  terms  in  the  second  members  except  the 
first,  and   Equations  (508)  become 


1.(.51-) 


§  305. — If  the  roots  p,  p,,  p.2,  &c.,  be  real,  the  different  terms  in  the 
values  of  ^,  tj,  ^,  <kc.,  as  given  in  Equations  (516),  will  disappear  period- 
ically, and  the  precise  times  of  disappearance  of  each  will  be  found  by 
making 

t  Vp  —  r  =  an;     t  Vp^  —  r,  =  a  tt  ;     t  -y/p^  —  r2  =  a7T]    <fec., 


or, 


t  = 


(iTC  -\-  r 
Vp 


ciTT  +  r,               an  +  r^ 
t  = -=  ;     t  = y=^  ;    &;c., 

Vp, 


P2 


in  which  a  is  any  Avhole  number  whatever.     The  intervals  of  disappear- 
ance will  be 


17  +  ?•         TT  +  r,         TT  +  7' 


Vp  Vp,  Vp 


= ;  <fec. 


MECIIANK'S     OF    MOLECULES.  351 

When  these  intervals  are  eumuiunsurable,  then  will  ^,  ?/,  ^,  «tc.,  resume 
the  values  they  had  at  some  previous  time,  the  molecules  will  i-eturn  to 
their  former  simultaneous  places,  the  movement  will  become  periodical, 
and  the  period  will  be  equal  to  the  least  common  multiple  of  the  above 
intervals.  This  phenomenon  of  periodical  returns  of  molecules  to  their 
initial  places,  is  called  the  -periodicity  of  molecular  condition. 

§  306. — From  Equations  (516)  it  is  apparent  that  each  and  every 
individual  of  a  system  of  molecules  in  which  the  connection  is  such  as 
to  leave  n  of  their  co-ordinates  independent,  may,  when  slightly  dis- 
tui'bed  from  rest  in  positions  of  stable  equilibrium,  assume  a  number  n 
of  oscillatory  movements,  and  that  all  or  any  number  of  these  may  take 
[ilace  simultaneously.  And  conversely,  whatever  be  the  initial  deiano-e- 
mcnt  of  such  a  system,  the  resulting  motions  of  each  molecule  may  bo 
resolved  into  n  or  less  tlian  n  simple  components  parallel  to  each  of 
any  three  rectangular  axes.  Here  we  have,  under  a  different  form,  the 
principle  of  the  coexistence  of  small  motions. 

§  307. — Again,  let  ^i,  t/,,  i^,,  (fee,  be  the  values  of  ^,  ?/,  ^,  (fee,  when 
the  system  is  in  motion  by  the  action  of  one  set  of  forces ;  ^2)  Vii  '52' 
ifec,  Avhen  under  the  action  of  another  set,  and  so  on — the  initial  con- 
dition being  determined  for  each  set  of  movements — then.  Equation 
(516)  being  linear,  will  the  resultant  values  of  ^,  ?/,  ^,  (fee,  be  given  by 

^  =  ,^  +  r,  +  f  3  +   (fee, 

?/  =?/,  +  ??,  +  7/3  +     (fee, 

^=^, +  C.  +  ^3+  «fec.; 
and  here  we  also  have  again  the  superposition  of  small  motions.     That 
is,  each  molecule  may  take  up  simultaneously  the  motions  due  to  each 
disturbing  cause  acting  separately  and  alone. 

§  308. — Equations  (513)  may  also  be  satisfied  by  making 

tVp  —  r 


7]  =  R.N  . 

V 

i;  =  R.N 


tVp  - 
tVp- 


353  ELEMENTS    OF    ANALYTICAL    MECHANICS, 

which  ffive 

o 

df       '  1  ' 

d'?  tA^-r^ 

and  these  substituted  in  Equations  (513),  give  Equations  (515),  with 
the  exceptions  of  the  signs  of  the  terms  which  are  indej^endent  of  p. 
But  with  this  solution  there  would  he  no  limit  to  the  increase  of 
f,  Tj,  (^,  f,  (fcc,  which  is  contrary  to  the  conditions  that  the  disturbances 
are  to  continue  small.  In  fact,  this  last  solution  supposes  the  molecules 
to  be  moved  from  positions  of  unatuhli-  cquiUhrium ;  the  other,  which 
is  the  case  of  nature,  from  stable  cquUihrinm. 


/fb 


§  309. — It  thus  appears  that  every  molecule,  subjected  to  certain 
conditions  of  aggregation,  may,  when  disturbed  from  its  place  of  rela- 
tive rest,  describe,  under  the  action  of  surrounding  molecules,  a  closed 
orbit.  The  disturbed  molecule  being  acted  upon  by  its  neighbors,  v.ill 
react  upon  the  latter,  and  cause  them,  in  turn,  to  take  up  their  appro- 
priate paths;  and  the  same  being  true  of  the  next  molecules  in  order 
of  distance,  the  disturbance  will  be  jjrogressive  and  in  all  directions. 
That  is,  an  initial  disturbance  of  a  molecule  at  one  time  and  place, 
becomes  a  cause  of  disturbance  of  another  molecule  at  another  time 
and  place.  While,  therefore,  any  molecule  .1,  is  travelling  over  its 
orbit,  the  disturbance  is  being  propagated  on  all  sides,  and  at  the  in- 
stant the  former  completes  its  circuit,  the  latter  will  have  reached  a 
anoiccnle  A^,  in  the  distance,  which  will  then,  for  the  first  time,  begin 
to  move  ;  and  the  molecules  A^  and  A-i  will,  thereafter,  ahvays  be  at 
the  same  relative  distance  from  their  respective  starting  points.  In  the 
same  way,  a  molecule  J3,  still  further  in  the  distance,  will  begin  its 
first  circuit  when  A^  begins  its  second  and  yi,  its  third,  and  so  on. 


MECHANICS    OF    MOLECULES.  353 

Between  the  molecules  A,  and  A.^,  as  also  between  A^  and  A3,  Sec, 
molecules  will  be  found  at  all  distances  from  their  starting  points  and 
moving  in  all  directions,  consistently  with  the  dimensions  and  shapes  of 
their  respective  orbits.  The  term  /)7ii(se  is  used  to  express  the  condi- 
tion of  a  molecule  with  i-espect  to  its  displacement  and  the  dircclion 
of  its  motion. 

Molecules  are  said  to  be  in  similar  j^hases,  when  moving  in  parallel 
orbital  elements  and  in  the  same  direction ;  and  in  ojyposite  phages, 
when  moving  in  parallel  orbital  elements  and  in  opposite  directions. 

The  particular  form  of  aggregation  assumed  by  the  molecules  be- 
tween the  nearest  two  concentric  surfaces  in  which  the  same  phases 
simultaneously  exist  throughout,  is  called  a  -wave. 

A  surface  which  contains  molecules  only  in  similar  phases,  is 
called  a  wave  front.  This  latter  term  is  generally,  though  not  al- 
ways, applied  to  the  surface  upon  which  the  molecules  are  just  begin- 
ning to  move.  The  velocity  of  a  wave  front  will  always  be  that  of 
disturbance  propagation.  A  zvave  lein/th  is  the  interval,  measured  in 
the  direction  of  wave  propagation,  between  two  consecutive  surfaces 
upon  which  the  molecules  have  similar  phases. 

WAVE    FUNCTION. 

§  310. — Denote  the  masses  of  the  molecules  by  ?n,  m',  etc.;  the  co- 
ordinates of 

m     by    X,  y,  2, 

m'      "     X  +  A  .r,      y  +  A  y,      z  +  Az, 
m"     "     :r  +  A.<     y-fAy',     z  +  Az\ 
&c.,  &c.,  &c.,  (fee, 

and  the  distance  between  any  two  molecules,  as  m  and  in\  by  r;  then 
will 


r  =  -v/a  x'  +  A7/  +  Az' (518) 

Let  f{r)  be  the  intensity  of  the  reciprocal  action  between  m  and  m' ; 
in  which  /  denotes  any  function  whatever.  This  reciprocal  action  will 
determine  the  elastic  force  of  the  body. 

23 


554 


ELEMENTS    OF    ANALYTICAL    MECHANICS. 


Before  the  system  is  disturbed,  there  will  be  no  inertia  developed, 
the  inertia  terms  in  Equations  (  A  )  will  disappear,  and  we  shall  have 
for  the  action  on  any  molecule  as  m, 


(519) 


Now  suppose  the  system  slightly  disturbed,  and  denote  the  displacement 
at  the  time  t  in  the  direction  of  the  axes  x,  ?/,  z,  respectively,  of 

m      by  I,  ?/,  C, 

m'      "  ^+A|,  v  +  ^V,  C+AC, 

m"     "  l+Af,  7j  +  Arj',  ^  +  A  ^', 
&c.,             &c.,             (fee,  &c. 

Then,  denoting  the  change  in  r  by  Ar,  Equation  (518)  becomes 


r  + A?•  =  ^/(A.^•  +  A^)2+(Ay  +  A7/)■^+(A2  + Ag^    .    (520) 

and  by  the  principle  of  the  superposition    of   small    motions,  Equations 
(  A  )  give  for  the  action  on  m, 


y 


™  •  -r4  =  2/(r  +  A  r)  . -^-^ 


..^  =./(.■  + A. •j.^'^^.r 


(Z*^ 


r  +  A  ?• 
A2  + A(^ 


(521) 


But 


».;^  =  J/(,-  +  Ar).    _.^^^. 


1  1       Aj-       Ar^ 

— ;— —  = r  H ^ <fec., 


/(r  +  A  r)  =f{r)  +  'i/-t)  A  r  +  &c.. 


whence,  neglecting  the  poAvers  of  Ar  higher  than  the  first, 


MECHANICS     OF    MOLECULES. 


355 


r  -\-  Ar 


(...H-.a=f-?+(^-^)-[-(-  +  -^)- 


Squaring  Equation  (520),  neglecting  the  squares  of  Ar,  A  |,  Atj, 
and  A^,  and  subtracting  the  square  of  Equation  (518)  from  the  result, 
we  find 

Aa;.A|  +  A?/.A?/+A£.A^ 


A  r  = 


Substituting  this  above,  and  making 

f(r) 


(p{r) 


i^ir) 


df{r)       f{r) 
r'^ .  d  r         r^ 


(522) 


(523) 


Equations  (521)  become 

d^  ^ 
m  .  ^  =  2|  9  (r) .  A  ?  + 1/^  (r)  {Ax  .A^  + Ay  .Ar]  +  Az  .  A^)  .Ax] 

m  .  ^  =  2 19  (r)  .  A?/ +  i/;  (r)  (A.T  .  A^  +  Ay  .  A  7y  4- As  .  A^)  .  Ay } 


^'? 
Je 


'L\^{l-).Al  +  ^{r){Ax.Al  +  Ay.Ari  +  Az.Al).Az] 


1(524) 


Performing  the  multiplication  as  indicated  in  the  last  term  of  the  sec- 
ond members,  there  will  result  terms  of  the  form, 

2  xp  (r)  .  A  ?/ ,  A  .r  .  A  y  ;  2  i/)  (r)  .  A  ^  .  A  .-r  .  A  £• ;  2  i/)  (r)  .  A  ^  .  A  a; .  A  y  ; 
S  i/)  (r)  ,  A  ?;  .  A  y  .  A  £■ ;       Si/)  (r)  .A^.As.Ay;       2i/>  (r)  .  A  f  .  A  .r  .  A  .-  ; 

and  it  may  be  shown  by  the  process  of  §  164,  to  prove  the  existence 
of  principal  axes,  that  the  co-ordinate  axes  may  be  so  taken  as  to  cause 
these  terms  to  vanish.  Assuming  the  axes  to  satisfy  these  conditions, 
Equations  (524)  become 


^^•Jl,  =:2{9(r)+1/>(r)A.^•'J.Ai^, 


d^r\ 


"*'^""^^'^^''^+'^^'"^'^^'^^''' 


>  . 


d^^ 


'''•  Ji^  =  "^^('■)+'^^''^'^''^^^• 


.    (525) 


\be 


ELEMENTS    OF    ANALYTICAL    MECHANICS. 


Making 

mp'  =  cp  (y)  +  ip  (r)  .  A  x"^, 
mp"  =  9  (r)  +  ip  (r) .  A  7/, 
m;y"=  9(?-)  +1/)  (?-).A.r 

Equations  (525)  take  the  form, 


(526) 


di' 


^2'  •  '-^  S) 


(527) 


An  initial  and  arbiti-ary  displacement  of  a  molecule  at  one  time 
and  place,  becomes,  througli  a  series  of  actions  and  I'cactions  of 
tlie  molecular  forces  alone,  the  cause  of  displacement  of  anotlier 
niolecnle,  at  another  time  and  place.  In  this  latter  displacement, 
Avhich  results  alone  from  the  molecular  foi'ces,  the  molecular  motions 
must  take  place  in  the  direction  of  least  molecular  resistance.  This 
direction  is  at  right  angles  to  that  of  wave  propagation ;  for,,  the  force 
which  resists  the  approach  of  any  two  strata  of  molecules  will  be  much 
greater  than  that  which  opposes  their  sliding  the  one  by  the  other. 
Indeed,  this  view  is  abundantly  confirmed'  by  many  of  the  phenomena 
that  result  from  wave  transmission ;  and  it  will  be  taken  for  granted, 
without  further  i-emark,  that  the  molecular  orbits  are  in  planes  at  right 
angles  to  the  direction  of  wave  propagation. 

§  311. — The  first  of  Equations  (52*7)  appertains,  therefore,  to  wave 
propagation  in  the  plane  »/ £-,  the  second  in  the  plane  x  z,  and  the  third 
in  the  plane  xy. 

The  integrations  of  Equations  (527)  are  given  by 

2  ~ 
^  =  a,  .  sin  ~j-  {V^.t  —  ?•,), 

o  — 


n 


«y  .  sm  -^  (  Fy .  i  -  ?v). 


^  ==  a, .  sin  ^  ( K  .  «  —  r,). 


(528) 


il  E C  II  A  N  I  C  S    OF    JI 0  L  E C  U  L  E S . 


iJoI 


In  which  V^,  F^,  aiul  V.  are  tlie  velocities  v/ith  which  the  disturbance 
is  propagated  in  directions  perpendicular  to  the  axes  .r,  y,  and  2,  re- 
spectively; Aj,  Ay,  and  X,  the  shortest  distances,  in  the  same  directions, 
between  the  places  of  rest  of  any  two  molecules  that  may  have  at  the 
same  instant  the  same  phase;  ?*j,,  r^,  and  r^  the  distances  of  any  mole- 
cule's place  of  rest  from  that  of  primitive  disturbance,  estimated  in  the 
same  directions.     This  being  understood,  we  liave  the  relations, 


,  _v/; 


=  vj- 


Make 


l/'  +  z']     ?v  =  V.^-i  +  2'^;     r,  =  V:c'  +  y 


^  TT  2  IT  2  7r  V 

_F— 72-  —V  —  n  •       — -F—  ??•^ 

Kj.   —   llx  )  -,        '  y  ">  )  -,        ''  z    "21 

A,  A„  A. 


277 


^V I 


=  z^'. ; 


(529) 


and  the  above  become 


f  =  a^  .  sin  (iij, .  <  —  /I'j. .  r^),  ^ 
7]  =  a,^.  sin  (n^  .  t  -  k^  .  r^), 
^  T=  a,,  sin  (??..  .  <  —  A-'^ .  r,). 


(53(0 


§  312. — To  show  that  these  are  the  solutions  of  Equations  (521),  it 
will  be  sufficient  to  prove  that  they  will  satisfy  those  equations  with 
real  values  for  n^,  n,j^  and  n^.  Differentiate  twice  with  respect  to  t, 
and  we  have  '^b  --  ^  ^"^  [^^'^  "  '^^•"^^3  '^>  ^"^ 


d'i; 


-  n' .  C- 


(531) 


Give  to  r^,  r^,  and  r^  the  increments  Ar^,  Aj-j,,  and  Ar,,  respectively  j 
the  corresponding  increments  of  f,  ?;,  and  ^  are  A  |,  A  ?;,  and  A  ^,  and 
Equations  (530)  become 


358  ELEMENTS     OF     ANALYTICAL     MEC II  AN  1  v"!  S. 

I  +  A  §^  =  a,  .  sin  (?«^  .  t  —  k^  .  i\  +  K.'^  r^), 
7]  -\-  :^7]  —  a,j.  sin  (»y  .t  —  ky.  r,j  -\-  k^.A  r^), 
^  +  A  ^  =  a, .  sin  [n^ .  t  —  k\  .  r^  -\-  k^  .  A  r,). 

Developing  the  second  members,  regarding  n^.t  —  k^.i\,  n^.t  —  k^ .  r^, 
and  n^.t  —  k^ .  9\  as  single  arcs ;  subtracting  Equations  (530)  in  order, 
replacing  1  —  cos  k^Ar^,  1  —  cos  k^Ar^,  and  1  —  cos  k^Ar^  by  their 
respective  values,  we  nnd  v      y  /»-  ^)t  *»  /^ 

A  ^  =  —  2  ^ .  sin'^  ^-^^ 4-  sin  (^"^  A  r^)  .  a,  cos  {n^.t  —  k^  .  r^), 

A  ?;  =  —  27],  sin^  ^  '^       ^^  +  sin  {ky  A  r^,)  .  a^,  cos  {ii^  ,  t  —  k^  .  r^),  I    (532) 
A  ^■—  —  2  ^ .  sin^  — +  sin  (k^  A  r^)  .  a^  cos  (h,  .t  —  k^  .  i\ 

Substituting  these  in  the  second  members  of  Etpations  (527),  we  have 
— ^  =  —  2  f .  2  «' .  sin= . -\-     S  ;/  .  sin  {/■  A  /■  )  •  «   •  cos  (»  -t  -  k  .  r  ),    I 

d"  !)  (^'     A  '^    )  i 

— -  =  -  2)7 .  2»".sin2  — ^^ ^      +      S;/'.sin(<l-  Ar).a   .  cos  (»    .  i  -  1-  .  ?■  ),    i-(53S) 

d--  C  (K  A  ^J 

— 5  =  -  2  5  .  S^y"  .  sin2 4-     2//"  .  sin  (.(-^  A  r^)  .  a^  .  cos  {n^.t  -  k^.  r^). 

Tn  the  state  of  equilibrium  of  the  molecules,  we  may  suppose 
their  masses  equal,  two  and  two,  and  symmetrically  disposed  on 
either  side  of  that  whose  mass  is  in.  Indeed,  this  is  the  most  general 
way  in  which  we  may  conceive  the  equilibrium  to  exist.  Then,  since 
for  every  positive  arc  k^ .  A  r^.  there  will  be  an  equal  negative  one,  we 
must  have 

2  p'    .  sin  {k^  .  A  r^)  .  a^  .  cos  (»,  .  t  —  k^  .  ?;,)  =  0,  "] 

2  p"  .  sin  [k^ .  A  r,j)  .  c^ .  cos  {n^ .  t  —  k„ .  r,)  =  0,   )>  .     .     (534) 

•  i 

Sy>"' .  sin  (/.-, .  A  1\)  .  a, .  cos  (», .  (  —  k, .  ?•,)  =  0,  J 

and  therefore, 


MECHANICS    OF    MOLECULES. 


359 


dt^  ^       ^  2       ' 


(535) 


whence,  Equations  (531)  aixl  (535), 


n,''  =  2  2  w   .  sin'' , 

n/^2  2/'.sin=-^2^\ 

,,,     .  „    ^^,  .  A  r, 
11,-  —  2  S^y".  sm^  — ■■■, 


(536) 


which  are.  Equations  (52G)  and  (522),  real  values  for  v^,  n^,  and  7i^. 

§  313. — Substituting  the  values  of  ??,,  ?^„,  «j,  and  k^,  k,^,  k^,  Equa- 
tions (529),  there  will  result,  after  nuiltiplying  the  first,  second,  and 
third  by  1  =  A  rj'  -^  A  r/  ;  1  =  A  ?•/  -^  A  ?•/  ;  1  =  A  r;  -^  A  ?•/,  re- 
spectively. 


77  A  r. 


F,=  =  iS/  .A?-, 


y/  =  i2y'.Ar/ 


(77  A  r^  V  2 
"AT/ 


77  A  r, 


(^^)' 


77  A  r. 


A, 


F.'=:lS/".Ar/. 


(537) 


360  ELEMENTS     OF    ANALYTICAL    MECHANICS. 

^1  TVAVE    SECTION.  •     >, «- / 

§  314. — Resuming  either  of  Equations  (528),  say  the  first,  viz.: 

2  77 

I  =  a^  sin  -^  {V^.t  —  r^), 

it  is  apppa'ent  that  if  t  be  made  constant  and  r^  variable,  so  as  to  reach 
in  succession  all  the  molecules  in  its  direction  between  the  limits 

V^  A  —  X„    and   V^  .t^ 

the    displacement    |   will    also    vary,    and    from    zero    to    zero,    passing 
between    these    limits    through    the    maximum  values  a,  and    minimum 
value   —  a^ ;    thus   deter- 
mining  the    curved    line 
C  Z*,    of     the     annexed 
figure,  to    be    the    locus 
of  the  corresponding  dis- 
placed   molecules,  of  which  the  places  of  rest  are   on  the  straight  line 
A  j5,  coincident  in  direction  with   the    line  r^  in    the    plane  y  z.      And 
it  is  also  apparent   that    if   the    above  value  of  t  receive  an   increment, 
making  the  time  equal  to  t\  and,  with  this  new  value  for  the  time,  i\ 
be  made  to  vary  between  the  limits 

F^.  i'  — A,,    and  V^  .  t\ 

the  locus  of  the  corresponding  displaced  molecules  will  be  found  to 
have  shifted  its  place  to  C  D\  in  the  direction  towards  Avhich  the  dis- 
turbance is  propagated. 

This  peculiar  arrangement  of  a  series  of  consecutive  molecules,  by 
which  the  latter  are  made  to  occupy  the  various  positious,  arranged  in 
the  order  of  continuity  about  their  places  of  rest,  is,  as  we  have  seen, 
§  305,  called  a  luave,  and  the  functions.  Equations  (528),  from  which  a 
section  of  the  waves  may  be  constructed,  are  called  wave  functions. 

WAVE    VELOCITY. 

§  315. — From  cither  of  Equations  (537),  say  the  first,  it  appears 
that  the  velocity  of  wave  propagation  depends  upon  the  ratio  between 


MECIIANIOS     OF    MOLECULES.  361 

the  arc  ■ — 7—^   and   its  sine.     If  the  distance  An,  between  the  mole- 

cules,  in,  the  direction  of  r^,  have  any  appreciable  vaUie  as  compared 
■with  the  wave  length  A^,  this  ratio  will  be  less  than  unity;  and  in 
proportion  as  the  wave  length  increases,  in  tlie  same  medinm,  will  the 
velocity  increase.  When  the  distance  A  r^  is  insignificant  in  compari- 
son Avith  the  wave  length  A^,,  the  ratio  of  the  sine  to  the  arc  will  be 
unity,  and  that  factor  will  cease  to  appear. 

§  316. — If  the  medium  be  homogeneous,  then  will 

2)'  =  2^"  =  2^'"  ;     A  r^  =  A  r^  r=  A  r, ; 

and,  therefore, 

T^  =  F,  =  r, . 

That  is,  the  velocity  will  be  the  same  in  all  directions.  Denote  this 
velocity  by  F;    we  may  w^rite 

.   „  TT  .  A  r 

5 — 

V'  =  II. • (538) 

•TT  .  A  r\2 


/  TT  .   A  ?'\2 


in  which  the  two  factors  that  compose  the  second  member  ha\e  siich 
average  values  as  to  give  a  product  equal  to  the  sum  of  the  products 
which  make  up  the  second  members  of  either  of  Equations  (537). 

Supposing,  in  addition  to  the  existence  of  homogeneity,  that  the  in- 
terval between  the  molecules  is  insignificant  in  regard  to  the  wave 
length,  the  last  factor  of  Equations  (53*7)  reduces  to  unity,  and  taking 
the  axis  x  i"ii  the  direction  of  the  velocity  to  be  estimated,  A  r  becomes 
A  a;,  and,  first  of  Equations  (53*7), 

F^  =  iSiy.(A.tf ; 

replacing  p'  by  its  value.  Equations  (52G)  and  (523), 

2m        L    r  \    dr       r^  r^   /  i 

The  distances  between  the  molecules  being  very  small,  the  term  of 
which  Ax^  is  a  factor  may  be  neglected  in  comparison  with  that  con- 
taining A  x^,  and  the  above  may  be  written 


362  ELEMENTS    OF    ANALYTICAL    MECHANICS. 

V  =  —  .:Efh-).~  .Ax. 
A  X 

Now,  f{r)  . is  the   component  of  tlie   elastic  force  exerted  betweer 

two  molecules  whose  distance  is  r.  in  the  direction  of  the  axis  x ;    and 

Ax  .        \ 

f{r)  .  —  .  Ajc    is    the    quantity    of    work    of    this    component    acting 

throuo-h  a  distance  A  x.     jNIakiua; 

Ax 
2/(>-).—  =2.,, 

we  may,  by  the  princijile  of  parallel  forces,  write 

^f{r).^.Ax^2e^x/, 


iu  which  e^  is  the  sum  of  the  component  molecular  forces  which  act  on 
one  side  of  the  molecule  ?h,  in  the  direction  of  the  axis  .r,  or,  which 
is  the  same  thing,  the  elastic  force  limited  to  a  single  molecule ;  and 
Xj  the  path  over  which  this  force  would  perform  an  amount  of  work 
eqiial  to  that  measured  by  the  first  member.     Substituting  this  above, 

Denote  by  i  the  number  of  molecules  in  a  unit  of  length,  and  multiply 
both  numerator  and  denominator  by  i^ ;    we  have 

but  ^^ .  e^  is  the  clastic  force  extended  to  a  unit  of  surface,  and  is  the 
measure  of  the  elastic  force  of  the  medium ;  call  this  e.  The  factor 
ix^  is  the  number  of  molecules  in  the  distance  ..t^  ;  call  this  h.  The 
denominator  i^m  is  the  quantity  of  matter  in  a  unit  of  volume,  which 
is  the  density;  call  this  A,  and  the  above  becomes 


=  \/p 


V=\/-.k        (539) 

Denote  by  c  the  ratio  which  the   contraction  produced  in  a  given  vol- 


MECHANICS     OF     MOLECULES, 


363 


ume  of  the  medium  by  the  pressure  of  a  standard  atmosphere  A,  bears 
to  the  volume  -nithout  any  external  pressure ;  then  will 


A       q  .D    .30  '"'*"• 
c  c 


(540) 


in  -which  'g  is  the  force  of  gravity  and  D ^^  the  density  of  mercury  at 
a  standard  temperature. 

In  the  case  of  gases,  c  is  sensibly  equal  to  unity ;  for  if  such  bodies 
■were  relieved  from  their  atmospheric  pressures  they  -would  expand  in- 
definitely, thus  making  their  increments  of  volumes  sensibly  equal  to 
the  volumes  they  would  ultimately  attain. 


RELATION    OF    WAVE    VELOCITY    TO    WAVE    LENGTH. 

§  317. — Denote  the  resultant  displacement,  of  which  ^,  ?/,  and  ^  are 
the  components,  by  tf ;  and  the  angles  which  C  makes  with  the  axes 
rr,  y,  and  0,  by  a,  /3,  and  y,  respectively ;  then  will 

1=0'.  cos  a  ;     ?;  =  (?.  cos  j3  ;     ^  =  a" .  cos  y  ; 

which,  substituted  in  the  second  members  of  Equations  (531),  give 


--  =   -.^.«.    .COS., 


df 


=  —  (j  .Uy  .  COS  /3, 
=  —  rf  .  n^  .  COS  y. 


(541) 


Squaring,  adding,  taking  square  root,  and  denoting  the  resultant  by  £„, 
we  have 


I'he  first  member  is  the  square  of  the  resultant  acceleration  due  to  the 
molecular  action  developed  by  the  displacement  C. 

Denote  by  a^,  /3^,  and  y^  the    angles    which    the    direction    of   this 


364: 


ELEMP]NTS     OF    ANALYTICAL     MECHANICS. 


resultant    makes    witli^4tlie    axes  .c,   y,   and   £-,   respectively;    and    by    ip 
the  inclination  of  this  direction  to  that  of  displacement.      Then  will 

cos  ip  =  cos  a  .  cos  a^  +  cos  /3  .  cos  (3^  +  cos  y  .  cos  y^  .     .  (54.3) 

The  components  of   the  acceleration,  in  the  directions  of   the  axes  ,r,  y, 
and  z,  are,  respectively, 

e^  cos  a^ ;     e,^  cos  [3^ ;     e„,  cos  y^  ; 
and,  therefore,  Equations  (541), 

E^  cos  a^  =  —  (f  .  »/ .  cos  «,  '       . 

£,„  cos  |3^  =  —  tf  .  ^^/  .  c:()s  /3, 
e„  cos  y^  =  —  cr .  »/ .  cos  y. 


Whence, 


C  .  n' .  cos  a 


cos  a,  =  — 


cos  /3^  =  — 


cos  y^ 


£, 

» 

rf 

?? 

2 

cos 

/3 

e, 

. 

? 

5* 

» 

2_ 

cos 

y 

(544) 


These,  in  Equation  (543),  give,  Equations  (531),       ;^ 

£,„  .  cos  ip  =:  —  ci  .  v.^  =  —  (f  .  (»/  .  cos-  a  -\-  n,f  .  cos^  [3  +  /?/  .  cos^  y)  ; 
and  replacing  n  ,  ii,.,  n,j,  and  w^,  by  their  values,  E(|uations  (529), 

-^,  =  -^  •  cos^  "  +  X^'s  •  cos'  ^  +  X^2  •  cos'  y. 

(T  *  y  - 

But,  because  the  number  of  waves,  in  a  unit  of  time,  arising  from  the 
components  of  a  co union  initial  disturbance  must  be  the  same,  the 
coefficients  of  the  circular  functions  above  must  be  equal,  and  hence. 


yi         yi  yi 

-T^a  =  -ri  (cos^  «  +  cos'  /3  4-  cos-  y)  =  -i 

A  A,  A. 


(545) 


Whence  the  wave  velocity  is  proportional  to  tlie  wave  length. 


MECHANICS    OF    MOLECULES, 


3G5 


^^ 


SURFACE    OF    1:LASTICITV. 


§  318. — Replacing,  in  Equations  (541),  %,  n^,  n^,  by  their  values 
m  Equations  (529),  multiplying  the  first  by  c.7:X^.m,  the  second 
by  c  .7T  Xy  .  m,  and  the  third  by  c  .tt  A/  .  ???,  we  have 

c  .  TT .  X^- .  m  .  -—.  —  —  (f  .c  .  47t\  m  VJ  .  cos  a,  *' 
a  t 


c  .TT  .  Xj  .  m 


df 


d'^ 


(f  .c.4:7T\mV/.cosP,    y  .    (546) 


c  .Tc  .X^  .m  .  -—  =  —  tf  ,  c  .  4  7r\  ??i  F/  .  cos  y. 
d  t 


Now,  TT  .  Xx,  TT  .  Xy^,  and  tt  .  X^^  are  the  projections  of  the  waves  arising 
from  the  component  displacements  f,  7],  and  ^,  on  the  planes  i/z,  x  z, 
and  X I/,  respectively;  and  if  every  mofecule  in  each  of  these  waves  had 
the  same  acceleration,  the  first  members  would  measure  the  elastic  forces 
exerted  over  these  projections  by  making  c  equal  to  unity.  These  are, 
however,  not  equal ;  but  if  c  denote  a  proper  fractional  coefficient,  and 
fij.,  e  ,  and  e^  the  actual  elastic  forces  in  the  three  waves,  we  may  write. 


(54V) 


Ex  =  —"'•?•  V'x^  .  cos  a,   ") 

e^  —  —  <f  .  g  .  V,f  .  cos  P,    y  .     •     •     • 

tj  =  —  (t .  g  .  V^\  cos  y. 

m  which  g  =  4  c  .  7r\  w.     Squaring,  adding,  taking  square  root  of  sum, 
and  denoting  the  resultant  bv  e  , 


f^  =  VeJ  +  £,/  +  £^'  =  (f.g.V  VJ  .  cos-  a  +  F/  .  cos'  /3  +  F' .  cos'  y  ; 

from  which  it  is  apparent  that  if  the  displacement  be  made  in  the 
direction  of  either  axis,  the  elastic  force  Avill  be  wholly  in  the  direction 
of  that  axis — a  property  possessed  by  these  particular  axes  in  conse- 
quence of  the  fact  that  they  were  assumed  in  directions  to  satisfy  the 
conditions  of  symmetry  in  molecular  arrangement,  which  caused  Equa- 
tions   (524)    to    reduce  to    Equations    (525).      The    directions    of   these 


366 


ELEMENTS     OF    ANALYTICAL    MECHANICS. 


special    axes   are  called   axes  of  elasticity/.      The   resultant    elastic    force 
will  not,  in  general,  act  in  the  direction  of  the  displacement. 

Denote  the  angfles  which  e  makes  with  the  axes  of  elasticitv  bv  a. 
8^,  and  y^,  and  the  angle  which  it  makes  with  the  displacement  by  i/), 
then   will 

cos  ip  =  cos  a  .  cos  a  J  +  cos  /3  .  cos  jS^  +  cos  y  .  cos  y^ , 
e   .  cos  a^  =z  e^  =  —  tf  .  f  .  F/  .  cos  a, 
£  .  cos  /3^  =  fij,  =  —  C  .  f  .  Fy^ .  cos  /3, 
e  .  cos  y,  =r  e,  =  —  (T .  c  .  F/  .  cos  X. 


Whence 


cos  a,  =  — 


cf .  g  ,  VJ' .  cos  a 


cos  (3 


cos  y^  =  — 


cf  .c  .  VJ  .  cos  3 

/  =  — ^ — ; — -^  y 


(f  .  g  .  V,^ .  cos  y 


(548) 


which  substituted  above,  give, 

e  .  cos  ijj  —  — (r.{'.F'=  —  cf .  g  (VJ .  cos^  a  +  VJ  .  cos'  /3  +  VJ  .  cos'  y) ; 

in  which.  F    is  the  velocity  perpendicular  to  the  displacement.     Making 

V^  =  V;      V,  =  a-      V^  =  b;      V,  =  c; 
we  have 

V=  V(?7cos^^T'&^~co?^+~?7cos^    .     .     .     (549) 

The  quantities  a,  b,  and  c  are  called  definite  axes  of  elasticity,  in  con- 
tradistinction to  axes  of  elasticity  which  merely  give  direction.  The 
surface  of  which  the  above  is  the  equation,  is  called  the  surfece  of 
elasticity.  The  value  of  F  will  measure  the  velocity  of  any  point  on 
the  wave  surface  in  a  direction  normal  to  the  displacement,  and  being- 
squared  and  multiplied  by  o" .  g  will  give  the  elasticity  developed  in 
the  direction  of  the  displacement  itself. 


MECHANICS     OF    MOLECULES.  367 

The  definite  axes  of  elasticity  are  the  geometrical  axes  of  figure  of 
the  surface  of  elasticity;  tlie  general  axes  of  elasticity  are  dirt-ctioDs 
parallel  to  these,  aud  drawn  from  any  point  in  the  medium  taken  at 
pleasure. 


WAVE   SURFACE. 

§  319. — This  is  the  locus  of  those  molecules  which  have,  simulta- 
neously, the  same  phase,  §  309 ;  and  whatever  this  phase  may  be, 
the  particular  surface  charactei'ized  by  it  will  be  concentric  with  that 
which  marks,  at  any  epoch,  the  exterior  limits  of  the  disturbance,  or 
upon  which  the  molecules  are  beginning-  to  participate  in  the  disturb- 
ance propagation. 

It  is  now  the  question  to  determine  the  equation  of  this  latter 
surface ;  for  this  purpose,  assume  the  origin  of  co-ordinates  at  the 
point  of  primitive  disturbance,  and  let 

Ix  +  7n?/  +  nz  =  V (550) 

be  the  equation  of  a  plane  tangent  to  the  wave  front  at  any  point, 
and  at  the  end  of  a  unit  of  time.  The  coefficients  I,  m,  and  n,  will 
be  the  cosines  of  the  angles  which  the  normal  to  this  plane  makes 
with  the  axes  a't/z,  respectively,  and  its  length  will  measure  the 
velocity  F,  of  wave  propagation  in  its  own  direction.  This  plane  must 
be  parallel  to  the  displacement  aud  its  normal  perpendicular  thereto; 
hence  '    ^  ' 

I  cos  a  -f  7?i  cos  j3  +  ?i  cos  y  =  0     .     .     .     .    (551) ; 
also 

cos'  a  +  cos^  (3  +  cos^  y  =  1      .     .     .     .     (552). 

Equations  (549),  (550),  (551),  and  (552)  must  exist  simultaneously 
for  real  values  of  the  cosines  of  a,  13,  and  y.  To  find  an  equation 
which  shall  express  this  condition,  square  Eq.  (549),  and  divide  it  by 
V^  •  cos"  a,  it  becomes 


8G8  ELEMENTS     OF     ANALYTICAL     MECHANICS. 


«'  +  ^ ;;-  +  f2 ^ 

L'OS     rt  COS-   1/  J  __„ 

= • (553)- 


divide  Eq.  (551)  by  cos  cc,  we  li:ive 


cos  (3    ,         cos  y        ^  ,-c^^ 

/  +  m ~  +  ?i ^  =  0 (554)  ; 

cos  u  COS  a 


and  divide  Eq.   (552)  by  cos'  a,  the  result  is 


,     ,    cos-/?        C0S2  y  1 

COS^  a  COS-  a  C0S2  a 


'  Equations  (553)  and  (555)  give 


"     I    o f-  c- 

COS-  /3  ens2  y  Co>-;  „  cos-  a 

COS-  «  Cos-  a.  V'^ 


whence 


r2_«24-(F2- 6-).-^-^  +  (F^>-c=).— ^  =  0         ....     (556). 

cos-  a  COS-  (1 

From  Equation  (554)  we  have 

COS  y  COS  a 

cos  a  n  ' 

which  in  Equation  (556)  gives 

COS"  P                                   Cos  j3 
UV-2  _ 7/),i24- ( V- - c')vi^'\ 1-2 kV"--  c"-)'l-  m =  -{V^-cC-)n"--{V"—c'^)l?, 

COS=  a  cos  u 

or 

cos2  p       ^  ( F^  _  c".)  .l.m  _  cos_^  _  _  (  T"-^  -  „.)  ^,2 -f  ( F^  -  c')  ^'  _ 

cos=«  "*"  ""  ( F-  -  h"-)  n"  +  ( F-i  -  c')  w^ '  cos  a~       (  F-  -  i-)  n=  +  {V^  -  c-)  »«= ' 

cos  (3     .  ...  . 

and  solvino-  with  respect  to  ,  tliere  will  result, 

'■  cos  a 


(557) ; 


MECHANICS    OF    MOLECULES.  369 

and  this  in  Equation   (554)  gives 

co>  Y _     (y^-l>-).n.l±m\/^iV^-(i^){V2-lj'i)n^+{V--a''){V^-c^)m^+(Vi-c^){y^-b^)li] 

(558). 

For  any  assumed  displacement,  ;Iic  v;,l;;c  of  V,  Eq.  (549),  becomes 
known,  and  the  vakies  of  the  first  members  of  Eqs.  (557)  and  (558) 
must  be  real;  whence  I,  in,  and  n,  must,  in  addition  to  Eq.  (549), 
also  satisfy  the  condition 

(  F^  _rt2)(  ya  _  jo),i2  -f  (  Fa  -  a^){V'  -  c-'ynf-  -\-  {Vi  -  b'^){V"-  -  e-)t-  =  0. 

Dividing-  by 

[V  -a')  {V  -}r)  {V  -c"-), 

and  inverting  the  order  of  the  terms, 

V                 m'  '             w'       _ 
V'-a^  +  y^in}  +  V'-c^  ~^ ^^^^' 

From  this  equation,  together  Avith  Equation  (550),  and  the  relation 

Z^  +  m^  +  «'  =  1, (560) 

we  have  all  the  conditions  necessary  to  find  the  equation  of  the  wave 
surface  ;  this  is  done  by  eliminating  F,  m,  /,  and  n. 

For  this  purpose,  differentiate    each   of  these  equations  with  respect 
to  the  quantities  to  be  eliminated.     We  have,  from  Equation  (550), 

(1) xdl  ■\- y  dm -{•  zdn  ^  dV  \ 

from  Equation  (560), 

(2) Idl  -\-  md  m  -\-  ndn  =:  0  '., 

and  from  Equation  (559), 

Idl       .     mdm  ndn    _  ^  ,       /        P  ,  nfl «« \ 

Multiply  the  first  by  A,  the  second  by  —  A',  the  third  by  —  1,  and 
add  members  to  members,  and  collect  the  coefficient*  of  like  "differ- 
ontials ;   there  will  result, 

24 


370 


ELEMENTS     OF    ANALYTICAL     MECHANICS. 


=  0. 


+  ^A y  -  A' m  -  -^'—^^d m 
+  {Xz  —  X'  n  —  — :,  j  d  n 


Tating  X  and  X'  of  such  values  as  to  make  the  coefficients  oi  dV  and 
dn  each  zero,  the  equation  will  reduce  to  the  first  two  terms;  and  as 
dm  and  dl  are  wholly  arbitrary,  Equation  (560),  as  long  as  dn  is 
undetermined,  we  may,  from  the  principle  of  indeterminate  coefficients, 
write, 

I 


(4)  .     .     .     . 

(5)  .     .     .     , 

(6) 

(7)  .    X-V 


Xx-X'l    -* 


V  -  a' 


0, 


Xy-X  m-  yT-JTi^  =  ^' 


Xz  -X'n  -y^—^=  0, 


:i7.-l- 


J^  + 


i^V'-dJ  '   {V'-bJ  '   {V  -c') 


=  0; 


Multiply  (4)  by  I,  (5)  by  m,   (6)   by  ?^,  add    and    reduce  by  Equations 
(550),  (5G0),  and  (559);    we  have 


(8) 


XV-X'=0\ 


■\rultiply  (4)  by  x,  (5)  by  y,  and  (6)  by  z;  add   and   reduce   by  Equa- 
tion (550)  and  the  relation  x'  +  ?/"  -{-  z-  =z  r' ;  we  have 


(a'F+w^  +  ^^^+     ''^ 


substituting-  for  ?J  its  value,  (8),  and  transposino-, 


o; 


V  -  c^ 


MECHANICS    OF    MOLECULES.  371 

transposing  in  (4),  (5),  and  (G),  squaring  and  adding,  we  have 


substituting  for  X'^  its  value,  (8),  and  reducing  by  (V),  we  have 

and,  therefore, 

(10)    ....     X=  y^^,,__  y^yi      ^'  =  ,..  _  yr 

Substituting  these  in  (4),  we  find 

F(?-^-   V)  ~      \7^~V'  "^   V  -  a)  ' 
X  '  VI 


whence 


similarly, 


y  V  m 

r^Zrb'  "^  V  -  b' ' 

z  F «    .  _ 

multiply  the  first  by  x,  the   second  by  y,  the  third   by  z,  add  and  re- 
duce by  (9)  and  (10) ;  we  have 

X'  if 


■/-  —  M-        T-  —  V        r  —  c^ 

From  this,  which  is  one  form  of  the  equation  of  the  wave  surface,  sub- 
tract 

x'  +  f  +  2'  _  , 
I2"  —  ^' 


and  we  have 


-, J  +  -r^>  + r  =  0    .    .    .    .    (o62) 

•^  —  a^        r^  —  b-        r-  —  & 


which  is  a  second  form  of  the  equation  of  the  wave  surface. 

Clearing  the  fractions,  it  becomes,  after  substituting  for  r^  its  value 
x'  +  f  +  z\ 


372 


ELEMENTS     OF     ANALYTICAL     MECHANICS, 


Ir 


b 


—  a^  (6-  +  r)  x' 
+  a'  ¥  e 

DOUBLE    "WAVE    VELOCITY. 


(5G3) 


§  320. — The  radius  vector  r  measures  the  velocity  of  the  point  (A 
the  wave  to  which  it  belongs;  and  denoting  by  ^^,  m^,  and  n^  the 
cosines  of  the  angles  which  r  makes  with  .r,  ?/,  and  2,  respectively, 
we  have 

X  r=.  T  .  l^\     y  =^  I'  ni^ ;     z  =z  rn^\ 

and   writing  F,  for  ?•,  we   have,  by  substituting  in  p](|Hation  (563),  and 
dividing  by  V^^ .  a^ .  b'^ .  c^, 

a  trinomial  equation,  of  which  the  second  powers  of  the  equal  roots  are 

and  in  which, 


A'  =1. 


^  _i 


'     /     1  _   1 
V       c'       a- 


+  «v 


yl"=  ^ 


V       c-        a'  V       c^       t? 


(566) 


(567) 


If  fl  >  6  >  c,  the  values  of  A'  and  yl"  will  be  real,  and  there  will,  in 
general,  be  two  real  values  for  -j^^;  and  with  this  condition.  Equation 
(565)  will    give    two  pairs  of  real  and  equal  roots  witli  contrary  signs. 


MECHANICS   OF   MOLECULES.  373 

The    positive   routs   give   two   velocities   in    any   one   direction,   and    the 
negative  in  a  direction  contrary  to  this. 

Through  the  origin,  conceive  two  lines  to  be  drawn,  making  -with 
the  axis  a,  angles  wliose  cosines  are  a^  and  a^^ ;  with  the  axis  b,  an- 
gles whose  cosines  are  /3^  and  /3^^ ;  with  the  axis  c,  angles  whose  co- 
sines are  y^  and  y^^ ;    and  such  that 


;    (5GS) 


and  denote  the  angle  which  r  makes  with   the  first  of  these    lines  by 
?<^,  and  that  which  it  makes  with  the  second  by  ti^^ ;    then  will 


c 

-?- 

J- 
c  *■ 

/ 

-    -■ 

/I     1 

\ 

1        1 

A'  =  Z,  «,  -f  n, .  y^  = 


cos  u 


l1 


A"  =  l^a.^-n^y^    =  cosu^,. 
Vl  —  A"  =  sin  u^ ;     VT^'A'^'  =  sin  u^^ . 
These,  in  Equation  (565),  give  for  the  two  values  of  -j— , 

'  r 

y-2  =  h  (7.  +  ^p)  +  I  (-i  -  -^  •  (cos  ^l, .  COS  u^,  -f  sin  u^ .  sin  u^^)  .  .  (5G9) 

y-2  =  ^  (^i  +  J')  +  ^?  ~  ^)  •  (^°' "'  •  '^^'^ ''"  ~  ^'"^ '''  •  '^"^ "")  •  •  (^ '  ^) 

and  by  subtraction, 

2     1 

Now, 

-y     and   Y 

are  the  retardations  of  wave  velocity.  As  long  as  a  and  c  differ,  the 
second  member  can  only  reduce  to  zero,  when  u^  or  tc^^  is  zero ;  whence 
it   appears  that,  as  a  general  rule,  every  direction   except  two  is  distin- 


374  ELEMENTS     OF     ANALYTICAL    MECHANICS. 

guislied  by  transmitting  two  waves,  one  in  advance  of  the  other.  The 
two  directions  which  form  the  exceptions  are  in  the  phiiie  of  the  axes 
of  greatest  and  least  elasticity,  and  make  with  these  axes  the  angles 
of  which  the  cosines  are  a^  and  y^,  a^^  and  y^^,  Equations  (568).  lu 
these  directions  the  waves  will  travel  with  equal  velocities. 

Any  direction  along  which  the  component  waves  travel  with  equal 
velocities  is  called  an  uxh  of  equal  wave  velocity.  All  bodies  in 
which  the  elasticities  in  three  rectangular  directions  difier,  possess, 
Equation  (o7l),  two  of  these  axes,  and  are  called  biaxial  bodies.  The 
I'etardation  of  one  component  wave  over  that  of  the  other,  will  vary 
Avith  the  inclination  of  the  direction  of  its  motion  to  the  axis  of  equal 
wave  velocity;  and  Equation  (571)  shows  that  the  loci  of  equal  retarda- 
tions will  be  arranged  in  the  form  of  siiherical  lemniscates  about  the 
poles  of  the  axes. 

§  321. — The  form  of  the  wave  surface    and    its    properties    become 
better  known  from  its  principal  sections  and  singular  points. 
Its  sections  by  the  planes  yz,  xz,  and  xy  give,  respectively, 


x  =  0;     if  +  z'-  a')  {b'f+c'z'  -  b'  c"-)  =  0, 

y  =  0  ;      iz^^x'-  h')  (r  ^  +  a'  x^  -  c' «')  =  0,  '^^ 

2  =  0;     (a;^  +  y^  -  c^)  {a?  x^  +h'y'''-  a-  b')  =  0, 


(572) 


If  a  be  greater,  and  c  less  than  &,  then  will  the  first  give  a  circle 
and  an  ellipse,  the  latter  lying  wholly  within  the  former ;  the  third 
will  give  the  same  kind  of  curves,  but  the  ellipse  will  wholly  envelop 
the  circle ;  the  second  will  give  the  same  kind  of  curves,  intersecting 
one  another  in  four  points.  This  last  is  the  most  important.  It  is  the 
section  ^;(/ra^/f^  to  the  axes  of  f/realeiit  and  least  elasticities. 

§  322.— If  b  =  c,  then,  Equations  (oC8), 


MECHANICS     OF    MOLECULES.  375 

the  axes  will  coincide  with  one  another  and  with  the  axis  a,  that  is, 
with  ic;    M^  will  equal  m^^,  and,  Equation  (571), 

1  1  /I  1  \       .  , 

Also,  Equation  (563), 

{x'  +  f  +  r  -  c')  [a'  .r  +  c"-  (/  +  z")  -  a?  c']  =  0  .     .     (5Y4) 

and  the  wave  surface  will  be  resolved  into  the  surface  of  a  sphere,  and 
that  of  an  ellipsoid  of  revolution.  Making  u^  =  0,  it  will  be  seen 
from  Equation  (571)  that  these  waves  travel  with  equal  velocities  in 
the  direction  of  the  axis  a.  For  any  other  value  for  it,  since  u  =  «  , 
cos  it^  cos  u^j  +  sin  u^  sin  u^^  =  1,  Equations  (5G9)  and  (570)  become 

and  it  hence  appears,  that  the  relocity  of  one  of  the  component  waves 
will  be  constant  throughout  its  entire  extent,  while  that  of  the  other 
will  be  variable  from  one  point  to  another.  The  first  is  called  the  or- 
dinary,  the  second  the  ext)xi-or dinar ij  tvare. 

If  c  be  greater  than  a,  then  will  the  ellipsoid  be  prolate ;  if  less 
than  a,  it  will  be  oblate.     There  is  but  one  direction   which  will    make 

K  ^  =  F.  I  and  that  is  coincident  with  the  axis  a.  Bodies  in  which 
this  is  true  have  but  one    axis  of   equal  wave  velocity,   and  are  called 

Uniaxial  bodies. 

From  Equation  (571)  it  appears  that  the  loci  of  equal  retardations 
are  concentric  circles,  of  Avhich  the  common  centre  is  on  the  axis  of 
equal  wave  velocity. 


UMBILIC    POINTS. 


§  323. — Let  X  =  0  represent  Equation  (5G3),  and  take 

1     dL  „        I     dL  _        \     dL 

cos  ^  = 7—  :     cos  B  — -r—  ;     cos  G  =  — ■  •  -J— ;        (o76) 

w      dz  10     dy  xo     dx 

in  whicli  J,  B,  and  C  are  the  angles  which  a  tangent  plane  to  the  sur 


376  ELEMENTS     OF     ANALYTICAL    MECHANICS. 


face    makes  with    the   co-ordinate    planes  xy,  xs,  and  j/ z,  respectively, 
and, 

1  1 


^m^ii^^i^  ■  ■  •  •  ^"'^ 


Performing  the  operation  here  indicated  on  Equation  (563),  we  have 
dL 


dz 

dJL 

dy 

d_L^ 
dx 


=  2z  (a^  x""  +  b'  y-  +  c'  z')  +  2  c' z  {x'  +  y-  +  z'  -  a^  —  b'), 

=  2y{a'  x'  +  6-  tf  +  c'  z')  +  2  b' y  (.c=  +  7/  +  z' -  a' —  c')  ; 

=  2  X  {a'  x'  +  b'y'  +  c  z')  +  2  a}  x  [x"  -f  y^  +  2^  -  6^  -  c^). 


Making   y  =  0,  brings  the   tangential  point    in    the    plane  a  c,  and   the 
above  become 


dL 

~dz 

dL 
dy 

dL 

dx 


=  2  z  {a'  x'  +  c'  z'}  +  2c'z  {x'  +  z'  -  a'  -  b% 

=  0, 

=  2x  {a'  x'  +  c-  z)  +  2  a'  x  (.c^  +  z"  -  h''  -  r). 


>    .     .     (578) 


the    second  of  which    shows    the    tangent    plane    to    be    normal  to  the 
plane  a  c. 

But  y  =  0  gives,  Equations  (572), 

a;2  +  2=  -  6^  =  0  ;     a'  x'  +  c'  s-  -  a' r  =  0, 

whence  we  have 


a;  =  dr  c 


a'  -  b' 


(579) 


for  the  co-ordinates  of  the  points  in  which  the  circle  and  ellipse  inter 


MECPIANICS    OF    MOLECULES. 


:T7 


sect,  and  wliicli  are  real    as   long-  as  a  y-  b  "^  c. 
Equations  (oTG),  (oVV),  and  (5V8),  ue  have 


cos  A 


-  ;    cos  Jj  =z  - 
0 '  0 


cos  C 


Substitutino-  these   in 


0 
0' 


hence  the  points  of  intersection  of  the  eUipse  and  circle  in  the  plane 
of  the  axis  a  c,  are  the  vertices  of  couoidal  cusps,  each  having  a  t;in- 
geut  cone.  If  a  line  be  drawn  tangent  both  to  the  ellipse  and  tlie 
circle  in  the  plane  a  c,  the  tangential  points  will  belong  to  the  cir- 
cumference of  a  circle  along  -which  a  plane  through  this  line  niav  be 
drawn  tangent  to  the  wave  surface.  This  circumference  is  in  fart  the 
margin  of  the  conoidal  or  umbilic  cusp,  determined  bj  the  surface  of 
the  tangent  cone  reaching  its  limit  by  becoming  a  plane  in  the  ti'iad- 
ual  increase  of  the  inclination  of  its  elements,  as  the  tangential  rir- 
cumference  recedes  from  the  cusp  ppint.  A  narrow  annular  })hii!e 
wave,  starting  from  this  circle,  will  contract  to  a  point  in  one  direc- 
tion ;  and,  conversely,  an  element  of  a  plane  wave  starting  in  the  op- 
posite direction  will  expand  into  a  ring. 

-  It  thus  appears  that  the  general  wave  surface,  and  of  which  (56.3) 
is  the  equation,  consists  of  two  napiies^  the  one  wholly  within  the 
other,  except  at  four  points,  where  they 
unite,  and  at  each  of  which  they  form 
a  double  umbilic,  somewhat  after  the 
manner  of  the  opposite  nappes  of  a.very 
obtuse  cone.  The  figure  represents  a 
model  of  the  wave  surface,  so  cut,  by 
three  rectangular  planes,  as  to  show  two 
of  the  umbilic  points,  as  well  as  the 
general  course  of  the  nappes,  by  the  re- 
moval of  a  pair  of  the  resulting  dicdral 
quadrantal  fragments. 


^7 


MOLECULAR   VELOCITY. 


§324. — Multiply  the  first  of  Equations  (531)  by  2  c?  ^,  the  second 
by  2  0? 7/,  the  third  by  2(Z^,  and  integrate;  there  will  result,  recollect- 
ing that  the  molecule  is  moved  from  its  place  of  rest, 


378  ELEMENTS     OF    ANALYTICAL    MECHANICS. 


d  P 


(580) 


whence  it  appears  that  the  velocity  of  a  molecule  in  the  direction  of 
either  axis  is  proportional  to  its  displacement  in  that  direction,  from 
its  place  of  rest.  The  place  of  rest  is  only  relative.  When  a  mole- 
cule is  in  a  position  such  that  its  neighbors  are  symmetrically  disposed 
around  it,  it  is  in  its  place  of  rest,  and  its  displacement  therefrom  will 
be  directly  proportional  to  the  excess  of  condensation  on  one  side  over 
that  on  the  other.  This  excess  and  the  molecule's  motion  will  reduce 
to  zero  simultaneously,  and  a  single  displacement,  not  repeated,  can 
only  give  rise  to  what  is  called  a  'pulse. 

These    equations    also    show    that   the    Uving  force  of  the  molecule  is 
'pro20O7-tional  to  the  square  of  th,e  displacement. 

MOLECULAR    ORBITS. 

§  325. — The  molecular  orbits  are  on  the  wave  front.  Suppose  the 
wave  due  to  the  displacement  g  to  be  superposed  upon  that  due  to  ?/, 
and  take  a  molecule  of  which  the  place  of  rest  is  on  the  axis  z. 
The  first  and  second  of  Equations*  (528),  will  be  sufficient  to  find  the 
orbit  of  this  molecule  under  the  simultaneous  action  of  both  waves. 
From  these  two  equations  we  find,  after  writing  z  for  r^  and  r,,. 


2  7r  .  -1  I 

(F,.^-.)-sm      - 


(1).     .     .     .     tl^.{V,.t-z)=^n 


2  TT       .^^  ,  .—it] 

(2).     .     .     .  (F,.i_.)=sm      -L. 


K 


(3) 


.     .     .     ^.(F..^-.)=eos-yi-i;, 


X 


(4).   .   .   .   i5.(r,.<-.)  =  cos-yi-l-. 


X 


MECHANICS     0¥    MOLECULES.  379 

Subtracting  (2)  from  (1), 


V^.t-z        V,.t-z\  ._.  f         ._,r, 

2  -  I r ^ r )   =   Sin SlU        — 


in  ■svliich  V^  .  t  —  z,  is  tlie  distance  of  the  wave  front  due  to  f  from 
the  molecule's  place  of  rest,  and  V^ .  t  —  z,  that  of  the  wave  front  due 
to  1]  from  the  same  point.     Make 

/j  =  time  required  for  the  wave  front  due  to  ^  to  travel  over  V^.  t  —  z; 

—    u  a  li  u  u  ■)    , 

'X  ~  A,  , 

t     —  "  "  "  7?  "  F      /  _  s  • 

—     U  U  ((  ((  li  •)     , 

'  „  —  A„  , 


then   will 

f  —   ■y  V        f   —   ^  t  i  f         T     —    i        T  t 

;     (581) 


V^A  —  Z  V„.t  —  Z_^t^  t,,   _   t,  .  T„  —  t„  .  T,    _     t^  ^ 


which  substituted  above  o-ives,  after  taking  cosine  of  both  members, 


r-  ri    ^  n  ^  n 


cos2^         '  ^'    ^  =         ../    n  ,        ,      =         ^ 


Clearino-  the  radical  and  reducing. 


2  cos  2  7T  ^-^  •  —  •  ^  -  sin'  2  tt  — ^  =  0  .     .     (582) 


which  is  the  equation  of  an  ellipse  referred  to  its  centre. 

§  326. — To  find  the  position  of  the  transverse  axis,  take  the  usual 
formulas  for  the  transformation  of  co-ordinates  from  one  set,  which  are 
rectangular,  to  another,  also  rectangular.     They  arc, 

l^  =  ^'  cos  tp  —  r]'  sin  9, 
?/  =  ^'  sin  9  +  T]'  cos  9  ; 

in  which  9  is  the  angle  which  the  axis  ^'  makes  with  that  of  ^. 

Substituting    these    values  of   f  and   ?/   in   "CquaticM  (582),  collecting 


3S0  ELEMENTS    OF    ANALYTICAL    MECHANICS. 

the  coefficients,  and  placing  that  of  the  rectauglc  ^'  ?;',  equal  to  zero, 
we  have 

2  sin  9  .  cos  9  («/  —  c./)  —  2  (siu^  9  —  cos'  9)  .  a,  .  a,, .  cos  2  -  -f  =  0  ; 

and  because 

sin"  9  —  cos^  9  r=  cos  2  9, 

2  sin  9  .  cos  9  =  sin  2  9, 

the  above  becomes, 

tt,  .  a„  t, 

tan  29  =  2.  —^ — ■—, .  cos  2  -  .  —     .     .     .     .     (583) 
<  —  «/  "/ 

R  327. Xow,  if  the  successive  pairs  of  component  waves  which  dis- 
turb   the    molecule,  reach  it  witli  a  variable    difference    of   phase,  then 

will   cos  2  77  —  be  variable,  and  the  transverse  axis  of   the  elliptical   or- 

bit  be  continually  shifting  its  place.  A  wave  in  which  the  molecular 
motions  fulfil  this  condition  is  called  a  common  wave;  being  far  the 
most  frequent  in  nature.  When  the  successive  pairs  of  component 
waves  are  such  as  to  make  the  second  member  of  Equation  (583)  con- 
stant, the  transverse  axes  of  the  molecular  orbits  will  retain  the  same 
direction,  and  the  w^ave  is  said  to  be  eUiptkullij  polarized. 

g  328. — If  —  equal  i,  or  any  odd  multiple  of  i    and  a^  =  a^  then 

will,  Equation  (582), 

f  +  /?=-«x'  =  0, (584) 

and  the  orbit  becomes  a  circle.  When  this  happens,  the  wave  is  said 
to  be  circularly  polarized. 

g  329. — If  -L  be  equal  to  any  even  multiple  of  i,  then  will 


cos  2  77 .  ^  =  1 ;      sm-  2  77  •  -^  z=  0  ; 


and,  Equation  (582), 


1-1  =  0, (585) 


MECHANICS    OF    MOLECULES.  3S1 

and  the  orbit  is  a  straight  line  through  the  molecule's  place  of  rest. 
The  motion  of  the  molecule  will  take  place  in  a  plane  normal  to  the 
wave  front,  and  the  wave  is  said  to  he  2^l<^'^^  polarized ;  and  a  plane 
normal  to  the  wave  front  and  in  the  molecular  paths,  is  called  the 
lilane  of  polarization. 

§  330. — Referring    the    curve    to    the    new    axes,   and   omitting   the 
accents  from  ^'  and  ?/',  Equation  (58*2)  may  be  written, 

^^         'rf         .  .  t 

—  +  —,-  sm^  2  -  .  -^  =  0, (586) 

m  which  a,  and  a„  "onll  take  new  values. 


REFLEXION    AXD    REFRACTIOX    OF    WAA'ES. 

§  331. — The  elastic  force  which  the  molecules  in  the  surface  of  one 
body  exert  upon  those  in  the  surface  of  another,  in  sensible  contact, 
must,  when  the  molecules  are  at  relative  rest,  be  equal  to  that  exerted 
by  the  molecules  in  the  interior  of  either  body ;  else  these  surface 
molecules  would  be  urged  in  opposite  directions  by  unequal  forces,  and 
relative  repose  would  be  impossible.  But,  for  oqual  displacements,  the 
elastic  forces  developed  in  different  bodies  are  in  general  unequal,  and 
this  is  one  of  the  most  common  of  the  causes  that  produce  a  resolu- 
tion of  primitive  into  secondary  or  component  waves. 

The  velocity  of  a  wave  molecule  varies,  Equations  (580),  directly  as 
the  molecule's  distance  from  its  place  of  rest.  If,  therefore,  a  wave,  in 
its  progress  through  any  medium,  meet  with  a  constitutional  change  of 
elasticity  or  density,  the  elastic  force  developed  at  the  place  of  change 
will  either  be  greater  or  less  than  that  which  determined  the  places  of 
rest  in  the  interior  of  either  body.  In  the  first  ease,  the  condensation 
in  front  cannot,  by  the  forward  movement,  reduce  to  an  equality  with 
that  behind ;  the  surface  molecules  will  first  be  checked,  and  then  purily 
driven  back  upon  those  behind,  and  a  return  and  an  onward  pulse  will 
proceed  in  opposite  directions  from  the  surface  which  marks  the  change 
of  structure,  as  from  a  primitive  disturbance.  In  the  second  case,  the 
moiecuies,   meeting   with   less  opposition,  will    go   beyond    their   '.u'-.ural 


OQi) 


ELEMENTS    OF    ANALYTICAL    MEOHANICS. 


limits  with  reference  to  those  behind,  the  hitter  will  close  up  in  sue 
cession,  and  thus  a  return  and  transmitted  pulse  will  arise  as  before, 
but  with  this  difference,  viz. :  in  the  latter  case,  the  molecular  motions 
in  the  return  pulse  will  continue  in  the  same  direction  as  before, 
whereas,  in  the  former  case,  those  motions  will  be  reversed.  The  return 
pulse  is  said  to  be  reflected ;  that  transmitted,  refracted.  The  primitive 
pulse,  and  of  which  these  are  the  components,  is  called  the  incident 
pulse.  A  change  of  density  or  of  elasticity  will,  Equation  (537),  pro- 
duce a  change  in  the  velocity  of  wave  propagation.  A  surface  which 
is  the  locus  of  a  change  of  density  or  of  elasticity,  is  called  a  deviating 
surface.  Two  planes  which  are  tangent,  the  one  to  the  deviating  sur- 
face, the  other  to  the  wave  front,  at  a  point  common  to  both,  will 
intersect  in  a  line  parallel  to  that  of  the  nodes  of  the  molecular  orbits, 
which  are  in  the  deviating  surface  and  near  the  common  tangential 
point.  This  line  of  intersection  is  called  the  line  of  nodes.  A  plane 
through  the  tangential  point  and  perpendicular  to  the  line  of  nodes,  is 
called  the  p/a??e  of  incidence.  The  medium  through  which  the  wave 
moves  before  it  meets  the  deviating  surface,  is  called  the  rnediuin  of 
incidence ;  that  into  which  it  enters  on  passing  this  surface,  the  medium 
of  intromittance. 


§  332. — Let  A  be  a  point 
common  both  to  the  wave  and 
deviating  surface.  A  C  &  lin- 
ear element  of  the  former,  and 
AB  a  like  element  of  the  lat- 
ter, both  lying  in  the  plane 
of  incidence.  Denote  by  V 
and  X  the  velocity  and  length 

of  the  wave  in  the  mediunr  of  incidence ;  by  F,  and  X^  the  same  in  that 
of  intromittance ;  and  by  t  the  time.  Now,  supposing  the  wave  to  proceed 
in  the  direction  C B,  and  taking  A  B  :=  d  s,  we  have  CB=:V.df. 

But  while  the  point  C,  in  the  incident  wave  front,  is  moving  from 
C  to  B,  the  reflected  pulse,  proceeding  from  ^  as  a  centre  of  disturb- 
ance, will  move  over  a  distance  equal  to  V  d  t  in  the  medium  of  inci- 
dence; the  refracted  pulse  over  a  distance  equal  to  V^.dt    in    that    of 


MECHAXICS     OF     MOLKCULES.  3S3 

iiitromittancc.     With  A  as  a  centre,  and  radius  V.  d  t,  describe  the  arc 

ac,  and  with  the  radius  V^d  t,  the  arc  a'  c' ',  and  from  B  draw  the  taiio-eiits 

B  D  and  B  D' ;  the  iirst  will  be  the  front  of  the  new  wave  element  in  the 

medium  of  incidence,  tlic  second  in  that  of  intromittance. 

§  333.— Denote  the  angle  C  A  B  =  A  B  D  by  9  ;  the  angle  .1  B  D'  by 

9';  then  will 

(/  s  .  sin  :-  =  Vd,  l\     ci?  s  .  sin  9'  =  F]  f/  <  .     ,     .     .     (oSV) 

and  by  division,  denoting  the  ratio  of  tlie  velocities  by  m, 

sin  CD         V  ,       . 

^—,  =17=»^ 588) 

sin  cp         Vj  ^       ' 

whence  sin  (p  =  m  sm  cp' (589) 

The  angle  9  measures  the  inclination  of  the  incident,  and  9'  that  of  the 
refracted  wave  to  the  deviating  surface.  These  are  equal,  respectively,  to 
the  angles  which  the  normals  to  the  incident  and  refracted  waves  make 
with  tlie  normal  to  the  deviating  surface,  at  the  })oint  of  incidence.  The 
first  is  called  the  anr/le  of  incidence,  the  second  the  (niglc  of  nf ruction. 
The  inclination  of  the  reflected  wave  to  the  deviating  surface,  is  called  the 
angle  of  reflexion.  The  normals  to  the  incident  and  reflected  waves  fall  on 
opposite  sides  of  the  normal  to  the  deviating  surface ;  and  because  the  ve- 
locity of  the  reflected  wave  is  equal  to  that  of  the  incident,  with  contrary 
sign.  Equation  (589)  becomes  applicable  to  the  reflected  wave,  by  making 
in  =  —  1. 

LIVING    FORCE    AND    QUANTITY    OF    MOTION    IN    A    PLANE    POLARIZED    WAVE. 

§  334.— Take  either  of  Eipiations  (528),  say  the  first,  and  which  I'clates 

to  a  wave  plane  polarized,  the  plane  of  polarization  being  perpendicular  to 

the  co-ordinate  plane  y  z,  diff"crcntiatc  with  respect  to  g  and  /,  dropping 

the  subscripts — we  get 

d^  2  cr  .  2  * 

'li=„.F.cos^(Fi-r    — . 

Denote  the  density  of  the  medium  by  \  and  the  area  of  any  portion 
of  the  wave-front  by  o,  then  will  the  mass  between  two  consecutive  posi- 
tions of  this  area  be  a.A.dr,  and  the  living  force  within  a  quarter  of  a 
wave-length  be 

-f-JA  '^^■-'   ^  r«-r  =  iA      ^  AAA  ^_^^^^ 

A .  iff  f 


384  ELEMENTS    OF     AXALYTICAL     MECHANICS. 

V 

Dividiu"'  bv  the  volume  « .  T",  and  reeallino-  tliat  c  and  —  are  constant, 

we  shall  find  that  the  quantity  of  living  force  in  a  unit  of  volume  of  the 
medium  will  vary  directly  as  the  product  of  the  density  and  square  of  the 
greatest  displacenjcnt ;  and  the  relation  of  these  products,  in  the  case  of 
any  two  waves,  will  determine  the  relation  of  the  effects  of  these  waves 
upon  the  organs  of  sense  upon  which  they  act. 

Again,  the  quantity  of  motion  in  this  quarter  of  wave-length  will  be 


A  .a.  d  )■.—-  =  /  A  ■  «  .  a  .  F.  cos  ^^(Vt  -  r)~—Jr=  a  -a  .  a  . 

+  ]y  at     J    Vl-r=^i\  A  A 


(591) 


RESOLUTION    OF    LIVING    FORCE    AND    OF    MOTION,    BY   DEVIATING    SURFACES. 

§  335. — Take  the  co-ordinate  plane  x z  in  the  plane  of  incidence,  and 
the  axis  z  in  the  direction  of  the  normal  to  the  incident  wave,  the  axis  y 
■will  be  parallel  to  the  line  of  the  nodes  <•!'  the  molecular  orbit  in  the  devi- 
ating surface,  at  the  place  of  incideni.-e.  Then,  pi'eserving  the  notation  of 
§  332,  will  the  element  of  the  deviating  sunace  at  the  place  of  incidence 
be  ds.dy,  and  its  projections  upon  the  incident,  reflected  and  refracted 
wave-fronts,  respectively,  be  ds  .dy .  cos^,  ds .  dy  cos(p,  and  ds  .dy.  cos  9'. 
These  will  take  the  place  of  f^  in  Equations  (590)  and  (591),  in  computing 
the  living  force  and  quantity  of  motion  in  the  incident,  reflected  and  re- 
fracted waves.  The  living  force  in  the  incident' must  be  equal  to  the  sum 
of  the  living  forces  of  its  reflected  and  refracted  components.  First  take 
the  wave  in  which  the  molecular  motions  are  parallel  to  the  axis  .r,  and 
employ  the  subscripts  /,  r  and  t  to  denote  the  incident,  reflected  and  re- 
fracted or  transmitted  waves,  respectively.  The  living  force  in  a  quarter 
of  each  of  these  waves  will,  omitting  the  common  factors.  Equations  (529), 
(545)  and  (590),  give 

A  .  cos  <p  .  V.  o}^,  +  A^ .  cos  <?' .  F^ .  a-^,  —  A  .  cos  9  .  F.  a^ji  =  0  ; 
or,  Equations  (588)  and  (589), 

,      ,   A^    cos  9'   sino'     .  „  ,       ^ 

A     cos  (p    sin  (p  ^       ' 

in  which  A  and  A^  are  the  densities  of  the  medium  of  incidence  and  of 
intromittance. 

The  molecular  motions  are  all  parallel  to  the  plane  of  incidence,  and  at 
the  same  time  normal  to  the  directions  of  their  respective  wave  motions ; 


MECHANICS     OF     MOLECULES.  385 

they,  therefore,  make  with  one  another  angles  equal  to  those  made  by  the 
directions  of  these  latter  motions,  and  we  obtain  two  more  equations  from 
the  relations  of  Equations  (59)  for  the  resokition  and  composition  of  ob- 
lique forces.  The  angles  made  by  the  direction  of  the  motion  in  the  inci- 
dent with  the  directions  of  the  motions  in  the  reflected  and  refracted 
waves,  are  180°  —  2:p  and  300°  —  (a;  —  <p'),  respectively ;  and  the  angles 
under  which  the  directions  of  the  motions  in  the  latter  waves  are  inclined 
to  one  another,  is  180°  —  (p-j-^'j.     "Whence 

rr  TT  sin  ((p  —  (p') 

A  .  cos  p  .  K  .  a,  _  =  —  A  .  cos  (p  .  K .  a, : .  - — ^^ t^  : 

^  ^  "   sm((p-f9')' 

,  rr  TT  sin  2  <p 

A^ .  coscp  .I- , .  a,,=  A  .  cosffl  .  V.  a, I .  - — ; ; 

^       '      ''  V  -    sm(9-l-(p') 

or,                                           sin  ((p  —  m')  /_«„\ 

'  a     -3  _  a       _J^!^ IJ        (593) 

sm(9-f9')  ^        ^ 

A     COS  o     sin©         sin  2  (B  z.^.x 

a^j  =  u^; . -.  •  -- — -  • ;- (o94) 

A^    cosip     siucp     sm  (9  -j-  9  ) 

Substituting  these  in  Equation  (592),  we  readily  find, 

A       4cos^<p'.  sin'(p' _  cos'<p'.  sin^cp' ^ 

A^  ~         sin'-  2  (p  cos^  9  .  sin**  9 

whence, 

/—         /-  sin  2  9         _     /-     cos  9.  sin  9  . 

VA^  =  Va-- ,-^. ^-VA. ,     .     ^,     .     .     (o95) 

'  2  cos  9.  sm  9  cos9.sm9 

Substituting  the  above  ratio  of  the  densities  in  the  equation  just  preced- 
ing, we  get 

2  cos  9'.  sin  9'  ,^     . 

sm(9-f9) 
multiplying  this  by  Equation  (595),  member  by  member,  and  the  equa- 
tion giving  the  value  of  »,,,  by  "v/a,  and  taking 

Va  .  a,;  =  1 ;      v'a  a^,  =  v\     Va^  .  a^^  =  w, 
we  find 

„=_!!ii(?^j?;i (59t) 

sm  (9  -f  9  ) 

«  =  ^^%-, ("98) 

sm  (9  4-  9') 

To  which  iijay  be  added  the  relations.  Equation  (589), 

sino  ,      ./,       sin'^  9 

sm  9  =  — ■- ;     cos  9  =  y  1 —r 

m  in 


386  ELEMENTS    OF    ANALYTICAL     MECHANICS. 

Transposing  the  term  of  wliicli  a^,  is  a  factor  to  the  second  member  in 
Equation  (592),  subtracting  Equation  (593)  from  c<^;  =  u^,-,  dividing  the 
first  result  by  the  second,  and  multiplying  the  quotient  by  Equation  (593)', 
we  readily  find 

"-ii±^-=-"-^ ^599) 

cos  (p  cos  (p 

That  is,  the  projection  in  the  direction  of  wave  propagation  and  on  the 
deviating  surface,  of  the  great(i^t  displacement  in  the  incident,  inei'eased 
by  that  in  the  reflected  Avave,  is  equal  to  like  projection  of  the  gi'eatest 
displacement  in  the  refracted  wave. 

Next,  take  the  wave  in  which  the  molecular  motions  are  parallel  to  the 
axisy;  these  are  parallel  to  the  deviating  surface.  The  motions  in  the 
incident,  reflected  and  refracted  Avaves  are  parallel  to  one  another,  and,  by 
the  principles  of  parallel  forces,  the  sum  of  the  motions  in  the  reflected 
and  refracted  waves  must  be  equal  to  that  in  the  incident.  The  equation 
for  the  living  force  will  be  the  same  as  before.  Whence  Equations  (529), 
(545)  and  (590),  omitting  the  common  factors, 

A  .  cos  (p  .  F.  u?yr  +  ^,  •  COS  9'.  V ^  .  a\t  —  A  .  cos  9  .  V.  a^yi  —  0  ; 

A  .  cos  cp  .  V.  a,,,  +  A^ .  cos  9'.  V^ .  c.j,,  —  A  .  cos  9  .  V.  u,,^  =z  0  .  (600) 
In  which  A  and  A^  are,  as  before,  the  densities  of  the  medium  of  incidence 
and  of  intromittance,  respectively;  or.  Equations  (588)  and  (589), 

A    sintp'   cos  9'     2  2 

'       A     sm  9    cos  9 

a'   sin  9'    cos  9'  ,       , 

a„A ^- r.a„,  — oc.  =  0      .      .     .      (COl) 

A    sm  9    cos  9      ■  • 

Transposing  the  terms  containing  a,,^  and  Wy^  to  the  second  members,  and 
dividing  the  first  by  the  second,  we  find 

«;+«,i  =  «,. (002) 

That  is,  the  greatest  displacement  in  the  refracted  is  equal  to  the  sum  of 
the  greatest  displacements  in  the  incident  and  reflected  waves. 

A'  .  .,        . 

Substituting  the  value  of  — ,  as  given  by  Equation  (597),  in  Equation 

(601),  we  have 

sin  9  .  cos  9  /^^„\ 

«vr+-T— V !^,-«,„-«,i=0.     .     .     .     (603) 

sm  9  .  cos  9 


MECHANICS    OF    MOLECULES.  3S7 

Substituting  in  this,  first  the  value  of  ciy^,  and  then  of  cc^^)  tleduccd  froui 
Equation  (602),  we  readily  get 

_  tan  (9  —  (p')  _ 


''*  tan  ((p  +  9') ' 
4  cos  (p'.  sin  9' 


....     (604) 

■^  •      sm  2  9  +  sm  2  9  ^       ' 

Multiplying  the  first  of  these  by  ^ ^,  and  the  second  by  Equation  (595), 
and  making 

there  ■will  result, 

,  _       tan  (9  —  9') 


tan  (9  +  ©') 
sin  2  (p 


ttj., .  cos  (p  cos  (p    sin  (9  —  9') 

a„,  sin  2  ©' 


....      (600) 

..  _    .     ,  „  ,  „    .     .     .     .     (60V) 

sin  (9  +  CO  )  .  cos  (9  —  <p  ) 

§  330.— Divide  Equation  (598)  by  Equation  (597),  and  Equation  (607) 

by  Equation  (606),  replace  «',  m,  v'  and  ?/'  by  their  values,  and  substitute  for 

the  ratio  of  the  square  roots  of  the  densities  its  value  as  given  in  Equation 

(595),  we  find 

a^n-costp'  cos 9'        sin  2  9' 

r 

-         .    .  ,x  1 r     •     •     •     (COS) 

(Xy,  sm  (9  —  9  )  .  cos  (9  +  9  ) 

But  a^^ .  cos  9'  and  a^, .  cos  9,  are  the  components  parallel  to  the  deviating 
surface  of  the  displacements  which  are  in  the  plane  of  incidence ;  a^^  and 
tty,  are  already  parallel  to  the  deviating  surface';  whence,  as  long  as  9>9', 
that  is,  as  long  as  the  velocity  of  wave-motion  in  the  medium  of  incidence 
exceeds  that  in  the  medium  of  intromittance,  the  molecular  phases  in  the 
refracted  and  reflected  waves  will  be  opposite,  and  conversely. 

§  337. — Denote  the  living  force  in  the  original  incident  wave,  sup- 
posed common,  by  unity  ;  that  in  each  of  its  two  original  components 
will  be  denoted  by  one  half  of  unity,  and  the  total  living  force  of  the 
reflected  wave  will.  Equations  (597),  (606),  be 

^2    gi„.  (^  ^  ^.|  -1    2    tan^  (9  +9')  ^       ^ 

and  that  of  the  refracted, 

u+u-^yi       sinM<P  +  9')/       '  \         tan-^(9  +  <PJ/     ^       ^ 


S'^S  ELEMENTS     OF    ANALYT    C  xV  L     MECHANICS. 


POLARIZAIIOX    BY    REFLEXION'    AND    REFKACTION. 

§  338. — The  first  term  in  the  second  member  of  Equation  (60  ), 
measures  the  living  foi'ce  in  that  portion  of  the  reflected  wave  wliich 
is  due  to  \ibratiuns  parallel  to  the  plane  of  incidence;  the  second,  that 
due  to  \ibrations  perpendicular  to  this  plane.  The  former  exceeds  the 
latter.  These  living  forces  being  [)roportionaI  to  the  squares  of  the 
UTL-atest  displacements,  the  former  may  be  represented    by  a/,  and   the 

latter   by  «,/,  in  Equation  (582).       The    factor  -^,  in  this  equation,  de- 

termines  the  ditference  of  phase  simultaneously  impressed  by  both 
waves  upon  the  same  molecule,  and  when  the  waves  have  passed  from 
one  medium  to  another,  its  value  will  depend  not  only  upon  the  na- 
ture of  both  media,  but  also  upon  the  action  to  which  the  waves  may 
have  been  subjected  while  crossing  the  space  wherein  the  physical 
chances  occur  that  constitute  the  transition  from  one  medium  to  an- 
other. The  amount  of  this  action,  in  any  particular  case,  can  only  be 
known  from  experience.  The  resultant  waves,  both  in  the  medium  of 
incidence  and  of  intromittance,  will  be  elliptically  polarized. 

When  9  -f  (p'  =  90°,  then,  Equation  (589),  will  sin  (p'  =  cos  9,  and 

sin  cp 
m= =^  tan  cp  ; ("1^) 

cos  (p  . 

the  second  term  of  Equation  (609)  will  disappear,  and  the  reflected 
Avave  will  be  wholly  polarized  in  the  plane  of  incidence.  This  angle, 
of  which  the  tangent  is  equal  to  the  index  of  refraction,  is  called  the 
iwlarizhig  anrjle. 

The  index  of  refraction  varies  with  the  wave  length,  Eqs.  (588).  (545), 
an<l  it  will,  tln-refore,  be  impossible  wholly  to  polarize,  by  a  single  re- 
flexion, a  wave  compounded  of  several  components,  having  difterent 
wave  lengths. 

Of  the  terms  of  the  second  member  of  Equation  (010),  the  last  is 
the  greater,  because 

sin-  (p  -  p')  _  tan'  ((p  —  ip')    cos^  (cp  —  cp')  ^ 
sin-  (.p  -f  cp')  ~  tan'^  (cp  -\-  cp')    cos'"  (9  -f  cp')  ' 


MECHANICS    OF    MOLECULES.  389 

and  the  excess  will  ineapurc  the  preponderance  of  that  part  of  the  i-e- 
liacted  wave  due  to  vibrations  perpendicular  over  that  due  to  vibra- 
tions parallel  to  the  plane  of  incidence.  This  excess  is  exactly  equal 
to  the  excess  in  the  reflected  wave  which  arises  from  vibrations  par- 
allel over  those  perpendicular  to  the  plane  of  incidence. 

§  338'.— If  the  wave  velocity  in  the  medium  of  incidence  be  less 
than  in  that  of  intromittance,  then  w'ill  m  be  less  than  unity,  and  the 
values  of  v  and  v'  become  imaginary  for  ail  angles  of  incidence  greater 
than  that  whose  sine  is  equal  to  ?«,  and  at  this  limit  the  problem 
changes  its  nature.  In  fact,  this  is  the  limit  of  reh-action,  acooniing 
to  the  law  of  the  sines,  Equation  (589),  and  for  any  increase  of  the 
angle  of  incidence  beyond  this,  the  wave  will  be  wjioliy    reflected. 

§  339. — If  the  wave  be  plane  polarized,  and  its  plane  of  polarization 
inclined  to  that  of  incidence,  under  any  angle  denoted  by  r/,  tlu-n  will 
the  reflected  component  displacements  parallel  and  perpendicular  to  the 
plane  of  incidence  be,  respectively.  Equations  (597)  and  (60G), 

ci"    Cm  —  rr^'\  taU    ((D  —  ffi') 

—  .  COS  a,  and -. — ; — -  .  sm  a. 


sin  (9  -h  ?') '  '  tan  (9  ■\-  9') 

The  component  waves  due  to  these  displacements  Mill  pi'oceed  onwards, 
and    may   satisfy  the    condition  of  -^   being    an    even    multiple    of   \ ; 

in  which  case  the  resultant  will,  Equation  (585),  be  a  plane  polari:a'd 
wave.  Denote  the  inclination  of  its  plane  of  polarization  to  that  of 
reflexion  by  a\  then  will 

.    tan  (?)  —  (p') 

tan  a'  =  ^-  =  ^^I^l^  ^  co^_^  _  ^^^  ^  ^    ^^^^^ 

V        sm  (3  —  (D  )  cos  (9  —  (p  ) 

^ — J- .  cos  a 

sm  {(p  +  cp'j 

If  (p  _j.  (p/  _  90°,  then  will  a'  =1  0°,  whatever  be  a  ;  also  if  a  =  0°, 
then  will  a'  =  0°  ;  finally,  if  9  =  0°,  then  will  9'  =  0,  and  a'  =  a. 
That  is,  when  a  plane  polarized  wave  is  incident  under  the  polarizing 
angle,  it  is  reflected  polarized  in  the  plane  of  reflexion.  Where  an  in- 
cident wave  is  polarized  in  the  plane  of   incidence,  the   reflected   wave 


300  EJ.EMENTS     OF     AIsALYTICAL     MECHANICS. 

preserves  its  plane  of  polarization  unchanged  under  all  angles  of  inci- 
dence. Finally,  under  a  perpendicular  incidence,  the  plane  of  polariza- 
tion of  the  incident  and  that  of  the  reflected  wave  coincide. 

■  Equation  (612)  shows  that  a'  is  always  less  than  a,  and  that  the 
plane  of  polarization  approaches  that  of  incidence  at  each  reflexio'n,  and 
may  be  made,  by  a  sufficient  number  of  reflexions,  ultimately  to  coin- 
cide with  it. 

§  340. — Still,  supposing  the  velocity  of  the  wave  less  in  the  medium 
of  incidence  than  in  that  of  intromittance,  or  (p'  >  9  ,  let  the  wave  be 
plane  polarized,  and  its  plane  of  polarization  inclined  to  that  of  inci- 
dence. The  vibrations  will  be  resolved  into  their  components,  respec- 
tively parallel  and  -perpendicular  to  this  latter   plane  ;    and    as   long    as 

sin  cp  <  ??i,  two  components  Avill  be  reflected  and  two  refracted.      If  — ^ 

'  / 

be  any  even  multiple  of  -j,  in  both  sets  of  components,  the  reflected 
and  intromitted  resultant  w-aves  will  be  plane  polarized. 

The  inclinations,  denoted  by  a'  and  a^,  of  the  planes  of  polariza- 
tion of  the  reflected  and  refracted  waves,  respectively,  to  the  plane  of 
incidence,  will  be  given  by 

,       v'  u'    ^ 

tan  a  =  —  •  tan  a  ;     tan  a.   =  —  .  tan  a  ; 

V  u 

in  which  z/,  v',  u  and  u',  are  to  be  found  by  Equations  (597),  (600), 
(598),  and  (60*7). 

If  a  =  45°  and  sin  9  =  ni,  then  will 

1 

tan  af  =  —  1,  and  tan  a^  =  — . 

m 

At  this  limit,  tlie  refiacted  wave  takes  the  direction  of  the  deviating 
surface.  An  infinitesimal  increment  to  9  will  canse  this  wave  to  hn 
reflected  and  make  ?/*  =  —  1,  tan  a^  =  —  1,  and  give  to  tan  a'  the 
form  of  indi'termination.  Ihit,  retaining  the  limiting  'alue  of  this  func- 
tion  above,  we   have, 

1  +  tan  a' .  tan  a^  =  1  —  1  zz:  0 ; 


MECHANICS     OF    MOLECULES.  391 

and  since  the  planes  of  polarization  pass  through  the  same  line,  viz., 
a  normal  to  the  wave  front,  they  will  make  with  one  another  an  ano-le 
of  90°,  and  the  whole  reflected  wave  will  be  compovnidcd  of  two  equal 
components  polarized  in  planes  at  right  angles  to  each  other.  If  these 
waves  reach  the  molecules    in    their  common  path,  so  as  to  satisfv  the 

condition  that  —  shall   be    an    even  multiple  of  A,   the   resultant   wave 

will  be  plane  polarized ;  if  an  odd  multiple,  then  circularly  polarized ; 
and  if  between  these  limits,  then  elliptically  polarized. 

§  341. — If  the  polarization  be  circular,  then  will  a,  =  a^,  =  a^,  be 
equal  to  the  radius  vector  of  the  circular  orbit.  Denote  the  angle 
which  this  radius  makes  with  the  axis  x,  at  any  instant,  by  6 ;  then 
will 

V^  .  t  —  z 
a^  .  cos  0  =  I'  =  a_j .  sin  2  77  — ^-^ , 

a,  .  sin  6  =  -n  =:a,, .  sin  2  tt  — ^ . 

Denote  the  time  required  for  the  first  wave  to  describe  V^ .  t  —  2',  by  t,, 
that  for  the  second  to  describe  V^.t  —  z  by  t^,  and  the  periodic  time 
of  a  molecule  in  both  waves  by  r ;  then,  because  the  wave  velocity 
is  constant,  and  the  wave  length  and  orbit  are  described  in  the  same 
time, 

V..t-z        L         V„.t-z        L 


"'i 


T  A„  T 


which,  in  the  above,  give. 


cos  d  —  sm  2  TT  .  — 


sin  0  =  sin  2  TT  .  — 


and  making 

t,=.t,±t\ (G13) 

in  which   t'  denotes    the    time    the    wave   due    to    vibrations    parallel   to 


392  ELEMENTS     OF    ANALYTICAL    MECHANICS. 

one  axis  is  iu  advance  of  that  due  to   tliDse  pdrallel  to  the  other;  Ava 
have, 

cos  0  =  sin  2  7T    -^  ; (C14) 


sin  0 


=:  sin  2  77    |--±— j.  .      ,     .      (615) 


Differentiating,  regarding  —  as  constant,  we  find, 


dd 

in 


^  2rr  /L    ,     t\ 

-  = ,  cos  2  7T  ( —  nz  —  I ; 

,       T.COS0  \t         r/' 


and,  developing  the  last  factor, 

dd            2  7T        r  t^  t'  .  t^       .  t'-i 

- —  = •  I  cos  2  TT  •  —  •  COS  2  TT  •  —  =p  Sin  2  7T  •  —  •  sin  2  77  — ■  I : 

dt^         -.COS  0      l  ■  T  T    ^  T  tJ' 

and  mating  —  =4-. 


T 


n      (^d  2  7-        .  t,  ,  , 

cos0.-p-  =  =F  ^3-.sin2  77.— (616) 

ci  tj.  T  T 

Differentiating  (614),  we  find, 

■     ^    (^0               2  77                   t,  ,      ^, 

sin0.-—  = ^— .cos2  77-£- (617) 

Squaring,  adding  to  the    square   of  Equation  (616),   and   taking  square 
root, 

dd  27T 

ur^- ("'"^ 

whence  the  velocity  is  constant. 

The  first  member  of  Equation  (616)  is  the  velocity  in  the  direction 
of  the  axis  y,  and  Equation  (617)  ih  the  direction  of  the  axis  .r,  and 
these  equations  show  that  the  upper  sign  must  be  taken  in  E(]uation 
(618)  when  t^  is  positive  in  Equation  (613),  and  the  lower  when  t'  is 
negative.  Whence  it  appears,  that  two  waves  plane  polarized  will,  by 
their  simultaneous  action  upon  a  molecule,  cause  it  to  move  uniformly 
in  a  circle,  provided  they  be  of  the  same  length,  and  one  wave  lag, 
as   it   were,  behind   the   other,   bv    a   distance  equal   to   |^  of  a   wave 


MECHANICS    OF    MOLECULES.  303 

lengtli ;  and  the  motion  will  be  from  right  to  left,  or  the  converse,  ac- 
cording to  wave  precedence. 

Two  waves  distinguished  by  these  peculiarities  are  said  to  be  ojipo- 
siteli/  polarized.  The  plane  perpendicular  to  the  wave  front,  and 
through  that  diameter  of  the  orbit  into  which  the  molecule  would  be 
brought  at  the  same  instant  by  the  separate  action  of  the  two  waves, 
is  called  the  plane  of  cro.'ising. 

§  342.— Let 
(1) a^  cos  6  =  ^  =  a,  sin  2  tt  — , 

(2) a^  sin  9  =  7/  =  a^  sin  l2~.  —  -] J,  ■; 

(3) a^  cos  0=^=0.^  sin  I  2  tt  .  -~  -\ V, 

(4) a,  sin  0  rr  7/  ^  ot/  sin  2  tt  — , 

be  the  displacements  in  two  oppositel}^  circularly  polarized  wases.  Tlie 
union  of  (1)  and  (4)  gives  a  resultant  wave  plane  polarized  ;  that  of 
(2)  and  (3)  also  a  wave  plane  polarized,  the  equation  of  the  path 
being 

in  the  plane  of  crossing.  It  thus  appears  that  the  union  of  two  circu- 
larly polarized  waves,  polarized  in  opposite  directions,  gives  a  j)lane 
polarized  wave,  of  which  the  intensity  is  double  of  either.  Conversely, 
a  wave  plane  polarized  may  be  resolved  into  two  components  of  equal 
intensity,  circularly  polarized  in  opposite  directions. 

§  343. — Because  the  time  of  describing  the  wave  length  is  equal  to 
the  molecular  periodic  time,  we  have,  denoting  the  velocity  of  wave 
propagation  by  V, 

X  =  Vr, 

whence 

X 


394:  ELEMENTS     OF     ANALYTICAL    MECHANICS. 

which,  in  Equation  (618),  gives,  after  multipying  by  t^  mid  dividing 
by  2  TT, 

cie 

dJ/  ^  __  Vt^ (619) 

The  first  member  is  the  arc,  expressed  in  circumferences,  described  by 
the  molecule  while  the  wave  is  moving  through  a  thickness  V .  ^  of 
the  medium.  So  that  a  wave,  compounded  of  many  components  hav- 
ing different  Avave  lengths,  but  all  polarized,  on  entering  a  medium, 
ma}^  emerge  with  the  planes  of  polarization  of  its  several  componenta 
so  twisted  through  different  angles  as  to  diverge  from  a  common  line 
perpendicular  to  the  wave  front.  The  department  of  optics  furnishes 
some  fine  examples  of  this.  A  piece  of  quartz,  of  a  peculiar  kind,  is 
known  to  twist  the  extreme  red  wave  through  an  angle  of  17°  29'  47'', 
and  the  extreme  violet,  44°  04'  58",  for  each  millimetre  of  thickness^ 

DIFFUSION    AND    DECAY    OF    LIVING    FORCE. 

§  344. — The  living  force  of  any  molecule  whose  mass  is  vi  and  ve- 
locity v^ ,  is 

m  v^ ; 

and  denoting  by  n  the  number  of  molecules  on  a  superficial  unit  of 
the  wave  front,  the  living  force  on  this  unit  will  be 

n  .  m  .  ?'/  ; 

and  on  the  surface  of  a  spheie  of  which  the  radius  is  r^, 

4  77  .  r/  ,  n  .  m  vf  ; 

and  for  another  sphere,  of  ■\^hich  the  radius  is  r^^,  and  molecular  velo- 
city v^,, 

4  TT  .  r^/  .  11  m  V ^f. 

If  these  spherical  surfaces  occupy  the  same  relative  positions  in  a  di- 
verging wave,  in  any  two  of  its  positions,  their  molecular  living  foicea 
must  be  equal ;  whence,  suppressing  the  common  factors, 

r/  .  m  vf  =  T^f  m  v^; (620) 


MECHANICS     OF     MOLECULES.  P,9o 

The  molecules  describe  elliptioal  crbits,  and  under  the  action  of  i::olcc- 
ular  forces  directed  to  the  centres  of  these  curves.  Tlie  periodic  tiiiio 
will,  therefore,  §  207,  Equation  (286),  be  constant,  however  the  dimen- 
sions of  these  orbits  may  vary ;  and  the  average  velocities  of  the  mole- 
cules will  be  proportional  to  the  lengths  of  their  respective  orbits,  or, 
in  similar  orbits,  to  any  homologous  dimensions  of  the  same — as  their 
transverse  axes  or  greatest  molecular  displacements.  Denoting  the  latter 
by  c'  and  c"  in  the  two  waves,  then  will 

v^         c' 

which,  with  Equation  (020),  gives 

c"  r^^  =  c'r^ (621) 

"VMience  it  appears,  that  the  living  force  of  the  molecules  of  any  wave 
varies  inversely  as  the  second,  and  the  greatest  displacement  inversely  as 
the  fij-st  2'^oiver  of  the  distance  to  which  the  wave  has  been  ijropaynted 
from  its  2^l«-ee  of  ijriiuitioe  disturbance. 

INTERFERENCE. 

§  345. — Resuming  Equation  (586),  viz., 

^  +  -4  -  siu^  2  77  -  =  0  ; 

denote  the  radius  vector  of  the  molecular  orbit  by  p',  and  the  angle  it 
makes  with  the  axis  of  |  by  6',  then  will 

^  =  p  .  cos  0' ;     ri  =  p' .  sin  6' ; 

which,  in  the  above,  give 

p  =  —  .  sm  2  7r .  - ; 

Va^/  cos'^  6'  +  «/  sin'^  6'  "^ 

and  mating 

«/  •  «//  _  y 


-/a^/.cos^e'-f  a/.sin^0' 


396  ELEMENTS     OF     ANALYTICAL    MECHANICS. 

Ave  have 

p'  =  c'  .  sin  2  ~  .  -        (622) 

In  this  equation,  p'  is  the  actual  disphiceuient  of  the  molecule  tVoni  its 
place  of  rest,  and  becomes  a  maximum   wlien  -  is  any  odd  mnltiple  of 

i      If,  however,  there  be  added  to  the  arc  2  -r  -,   an    arbitrary   arc   «', 

4  •  "  '  f 

tliis  latter  may  be  so  taken  as  to  make  the  maximum  or  any  other 
displacemi.'Ut  occur  at  such  time  and  place  as  we  please,  and,  there- 
ibre,  to  give  to  the  molecule  any  particular  phase  at  pleasu'/e,  at  the 
time  t.     We  may  write,  then,  generally, 

p'  =  c' .  sin  (2  TT  .  ^  +  «')  ;      .     .     .  .     (623) 

and  for  a  second  resultant  wave, 

t 


"  =c".sin(2~.  -  +  a")  ; (624) 


P 


and  if  these  waves  act  simultaneously  upon  the  same  molecules,  the  re- 
sultant displacement,  denoted  by  p,  will,   g  ^UO,  be  given  by 

p  =  p'  +  p"  =  c' .  sin  (2  TT  .  -  +  a')  +  r"  .  sin  ^2  tt  .  -  +  a"). 

Developino-    the    circular    functions   ard    collecting  the   coefllcients  of 
like  factors, 

p  =  (c'  cos  a'  +  c"  cos  a") .  sin  2  tt  -  +  (  '  sin  a'  +  c"  sin  a")  .  cos  2  tt  -  ; 

and  making 

c  cos  a  —  c'  .  cos  a'  -f  c"  cos  a", ) 

[ (G25) 

c  sin  a  —  c'  sin  a'  +  c"  sin  a", 


we  have 


^  .  t 

0  =^  c  .  cos  a  .  sm  2  tt  ,  — 1-  c  sin  a  .  cos  2  tt  .  - ; 
"^  T  T 


MECHANICS    OF    MOLECULES. 


or. 


•     /  i  \ 

p   =z   C   Sill    f  2  TT  ,  -   +  «/. 


Squaring  Equations  (025),  and  adding, 

c'  =  c"  +  c"-  +  2  c'  c"  cos  {a'  —  «"), 
and  dividing  the  second  by  the  first, 

c' .  sin  a'  +  c"  .  sin  a" 


tan  a 


c'  cos  a'  +  c"  cos  a' 


(G26) 


(627) 


(628) 


From  Equation  (626)  we  see  that  the  resultant  wave  is  of  the  same 
length  as  that  of  the  component  waves  to  which  Equations  (623)  and 
(624)  appertain;  the  length  being  determined  by  the  molecular  periodic 
time  T ;  but  the  value  of  a  in  that  equation  differing  fi'ora  a'  and  a" 
in  Equations  (623)  and  (624),  shows  that  the  maximum  displacement  of 
a  given  molecule  does  not  take  place  in  the  resultant  wave  at  the  same 
time  as  in  either  of  its  components. 

§  346. — The  maximum  displacement  in  the  resultant  wave  is  given  bv 

c  =  Vc"  +  c'"  +  2  c'  c"  .  cos  {a'  -  a")  ;    .     .     .     (029) 

which  will  be  the  greatest  possible  when  a'  —  a"  =  0,  and  least  pos- 
sible when  a'  —  a"  =  180°  ;  the  maximum  in  the  former  case  being 
given  by 

c  =  c'  +  c" 
and  the  minimum,  by 


In  the  first  case,  Equation  (628), 

(c'  +  c")  .  sin  a' 

tan  a  =  f-, ;- =  tan  a  . 

(c  +  c  )  .  cos  a^ 

Whence  a  =  a'  =  a",  and    the    maximum    displacement  will    occur    at 
the  same  place  and  time  in  the  resultant  and  component  waves. 


39S  ELEMENTS     OF    ANALYTICAL     MECHANICS. 

In  the  second  case,  Equation  (628),  if  we  make  a    =  1S;1°  +  a", 

(f'  —  c")  .  sin  a" 

tan  a  = -■{ =  tan  a     =  tan  (a   —  180  )  =  tan  a  ; 

(c  —  f  )  .  cos  « 

that  is,  a  will  be  equal  to  one  at  least  of  the  arcs  a!  and  a",  and  the 
greatest  displacement  will    occur    at    the    same    time    and    place  in  the 
resultant  wave  as  in  one  of  its  components. 
If  c'  —  c",  then,  Equation  (629), 

c  =  c'  v/2  [1  +  cos  (a'  — a")]  ; 
and  because 

«'  —  a" 
1  +  cos  {al  —  a")  =  2  cos^ — , 


c  =  2  c' .  cos , (630) 


and,  Equation  (628), 


sm  a   +  sin  a  a   -\-  a 

tan  a  = ; -r.  =  tan ....     (631) 

cos  a   +  cos  a  2 

If,  while  c'  and  c"  continue  equal,  we  also  have  a'  —  a"  =  180°,  then. 

Equation   (630), 

c  =  0. 

Thus  it  appears  that  two  equal  waves  may  reach  the  same  molecules 
in  such  relative  condition  as  to  keep  tliem  in  their  plac(;s  of  rest ;  in 
other  words,  two  equal  waves  may  destroy  one  another. 

O  §  34*7. — To  ascertain  the  precise   relation   of  two    waves   which    will 

cause  this  mutual  destruction,  make,  in  Equation  (623), 

2  7T  .  T 

a'  =  a"±7T  =  a"  ±  -— ^, 

2  T 

and  that  equation  becomes, 

,         .       /  i  „     ■     27T.T\ 

p'  =  c'.sm^27r-  +  «"±^— J, 

(<  ±  It  \ 

2  77 ^ h«"l;      ....      (632) 


MECHANICS     OF     MOLECULES.  399 

which  becomes  ideutical  with  Equation  (624)   by  making 

c'  =  c", 
and 

t=tdzlT (633) 

Now,  the  same  vah;e  for  f,  in  Equations  (623)  and  (624),  will,  for 
equal  vahies  of  the  arbitrary  arcs  a'  and  a",  determine  the  component 
waves  to  give  to  a  molecule  subjected  to  their  simultaneous  action, 
similar  phases  ;  and  a  value  for  t,  in  the  one,  which  differs  from  that 
in  the  other,  by  one-half,  or  any  odd  multiple  of  one-half,  of  the 
molecular  periodic  time,  opposite  2'>hases.  And,  because  the  waves  pro- 
gress by  a  wave  length  during  each  molecular  revolution,  the  above 
result  shows  that,  when  two  loaves  meet,  after  having  travelled  over 
routes,  estimated  from'  points  at  which  their  molecular  phases  are  simi- 
lar, and  which  routes  differ  by  half,  or  any  odd  multiple  of  half  a 
wave  lenyth,  they  loill  destroy  one  another,  provided  the  waves  hare  the 
same  length  and  equal  maximum  molecular  dispilacements.  This  act,  by 
which  one  wave  destroys  another,  is  called  wave  interference. 

The    same    process    of   combination   will    equally  apply   to   three   or 
more  wave  functions  in  which  r  is  the  same  in  all ;  that  is,  wherein  the 

t  t 

wave  lengths  are  the  same  ;  for,  in  that  case,  sin  2  tt  .  -    and    cos  2  tt  ,  - 

being  common  factors,  after  developing  each  function  in  the  sum,  the 
resultant  displacement  p  becomes, 

p  =  sin  2  TT  .  -  .  2  c'  cos  a'  +  cos  2  7r  .  -  .  2  c'  sin  a', 

'^  T  T 

and  assuming 

c  .  cos  a  =  2  c'  cos  a', 
c  .  sin  ci  =  2  c'  sin  a' ; 

p  =  c.sin(2-n-- -f-a),    .     .     (634) 

thus  making  the  resultant  wave  of  the   same  length    as   that  of  either 

of  its  components. 


400  ELEMENTS     OF     ANALYTICAL    MECHANICS. 

But,  if  the    component  waves   be    not  of   equal    lengths,  the  sum   of 
the    corresponding    functions    cannot    reduce    to    the    form  of   Equation 
(634),  because   of  the 
absence     of     common  '    ^-\  ,''' — "^ 

factors,  arising  from  a        ^'  — -^^ \\- ^'^^z:;^ — ^V^ ^^'^ 

change    in    the    value  '^v y 

of  r   from    one    com- 
ponent to  another.     Such  components  can  never  destroy  one  another. 

INFLEXION. 

§  348. — Make,    in   Equation  (021),    ?•"  =  1,    and    that    equation    be- 
comes 

r  ^ 

and  this  value  being  substituted  for  c',  in  Equation  (622j,  gives, 

c"  t 

p'  =  —  .  sin  2  TT  ,  - : 
r  T 

and  making 

^_  Vt  —  r^ 

we  have,  omitting  all  the  accents, 

c  Yt r 

p  =  -  .  sin  2  71 , (635) 


r 


X      ' 


which  is  of  the  same  form  as  Equations  (528),  and  in  which  V  is 
the  velocity  of  Avave  propagation;  /,  the  time  of  its  motion  from 
primitive  disturbance ;  A,  the  wave  length  ;  -,  the  maximum  displace- 
ment of  a  molecule  of  which  the  distance  of  the  place  of  rest  from 
the  point  of  primitive  disturbance  is  r ;  and  p  the  actual  displacement, 
at  the  time  t,  of  this  same  molecule.  And  from  which  it  is  apparent 
that  the  displacements  will  always  be  the  same  for  equal  distances, 
V  t  —  r,  behind  the  Avavc  front. 

Every  disturbance  of  a   molecule,  at  one  time,  becomes   a   cause  of 


MECHANICS     OF    MOLECULES.  401 

dislurbauce  to  another  molecule  at  some  subsequent  time.  All  iho 
molecules  in  a  wave  front,  Avlien  they  first  bog-in  to  move,  become, 
therefore,  centres  of  disturbance  for  every  molecule  in  advance ;  and 
if  the  primitive  disturbance  be  kept  np,  secondary  waves  proceeding 
from  these  centres  will  reach  a  molecule  in  advance  simultaneously, 
and  determine,  §  307,  at  any  instant  <,  its  displacement  2  p. 

Suppose  a  wave,  whose 
centre  of  disturbance  is  C,  to 
have  reached  the  position  AB, 
so  remote  from  C  that  a  small 
portion,  yli>,  may  be  regarded 
as    sensibly   plane :     What    is 

the  displacement  of  a  molecule  at  0,  produced  by  the  simnltasHO-.is 
action  of  the  secondary  waves  proceeding  from  the  molecules  in  any 
portion,  as -^i>,  of  a  section  of  this  wave  front?  Draw  the  nornud 
CDJV,  through  the  middle  o(  P  Q  \  denote  the  variable  distance  J)  Q 
by  z,  and  Q  0  hj  r.  The  displacement  of  the  molecule  0,  by  the 
secondary  waves  from  the  arc  AB  =  2  h,  will,  Eq.  (635),  be  given  by 

l^p^J       pdz^j         ^.sin27r.-l-r^     .     .     (636) 

Here  r  and  z  are  variable.  To  eliminate  the  former,  join  0  with  the 
middle  of  AB  by  the  line  D  0,  and  denote  its  length  by  /,  and  the 
angle    Q  D  0,  which  it  makes  with  the  wave  front,  by  0.      Then   will 


r  =  Vl'  +  s'  —  2lz  cos  d  ; 
and  by  ]\Iaclaurin's  formula, 

sin^  0      5        n  /ro^\ 

r  =  l-cos6.z-]---y.z'-&c (63/) 

If  the  greatest  value  of  z  be  small  as  compared  to  I,  we  may  take 

r  =  1  —  cos  d.z,  (638) 

and  regard  the  displacements  of  the  molecule   0,  by  the  partial  waves 

26 


402  ELEMENTS     OF    ANALYTICAL    MECHANICS. 

from  z  to  be  equal.  Whence,  substituting  the  value  of  r,  with  this 
restriction,  in  Equation  {^'^'o)^  we  have, 

p  .dz  =  -  .  f        sin  -^  [Vt  —  I  -i-cos6  .z)  dz, 

and,  performing  the  integration  without  regard  to  limits, 

C  X  2  77 

2  p  = .  cos  ^—  (  K  i  —  /  +  cos  t^ .  z\ 

^  2  71/ cos  0  A    ^  ^' 

and  between  the  limits  —  h  and   +  ^; 

Sp  = -^ -^  '  \cos^-^  (Vt— l-h  co^e)—cos^  (Vt—l-{-b  .  cos  0)1, 

2  7T .1  .cosO    L        A  ^  A   ^  ^ -1 

or, 

cA             .    2  7r.i.cos0      .              F^  —  /  ,       , 

2p  = r ^  .  sm , .  sm  2  77 — - ;      ....     (639) 

77  .  /  .  COS  d  A,  A  ^  ' 

so  that  the  function  whose  value  gives  the  resultant  displacement,  is  of 
the  same  form  as  that  of  the  function  which  determines  either  of  the 
partial  displacements. 

The  maximum  value  of  the  resultant  displacement  is  given  by 

„  c  A  .    2  77 .  i .  cos  6  , 

2  p  =  —~ •  sm ;        ...     (640) 

rr .  l .  cosO  A  ^        ^ 

and  this  "will  become  zero  for  such  values  of  6  as  make  b  .  cos  0  equal 
to  either  of  the  following  values,  viz., 

i  A,     f  A,     I  A,     I  A,   &c. 

Conceiving  the  figure  to  be  revolved  about  the  normal  CJV,  and  all 
the  wave  except  the  circular  portion  whose  diameter  is  2  b  =  A  B,  to 
be  intercepted,  the  space  in  advance  of  the  wave  will,  when  the  above 
values  obtain,  find  itself  divided  by  the  secondary  waves  into  a  series 
of  concentric  cone-like  zones  around  the  normal  CJV,  as  an  axis,  and 
of  which  the  alternate  ones,  beginning  with  that  immediately  about  the 
axis,  Avill  be  filled  with  molecules  in  motion,  while  the  molecules  in  the 


MECHANICS     OF    MOLECULES.  403 

Others  will  be  at  rest.  A  section  in  advance  of  the  primitive  wave 
will  cut  from  these  zones  a  series  of  concentric  circular  rings  distiu- 
guisLed  by  the  same  peculiarities. 

But  if  X  be  very  great  as  compared  with  6,  then  will  the  arc 

2  ~  .  i  .  cos  0 


be  so  small  as  to  justify  the  substitution  of  the  arc  for  its  sine  and  for 
the  maximum  value  of  resultant  displacement, 

,^    .  cX  2  TT .  i  .  cos  0       2cb 

^"^p^'-^^jTVoTo X =  -/-'•   •   •    (6*1) 

and  this  result  being  independent  of  d,  the  conic  zones  cannot  exist, 
and  the  effect  of  the  secondary  waves  will  be  diffused  in  all  directions 
to  the  front.  This  lateral  action  of  secondary  Avaves  proceeding  from 
a  small  portion  of  a  primitive  wave,  is  called  wave  wjlection. 

When  d  approaches   nearly  to  90°,  cos  6  will  be  exceedingly  small, 
and  the  arc 

2  7r  .  i  .  cos  d 


X 

may  again  be  substituted   for    its  sine ;    again  Equation  (641)  suits  the 

case,  and  determines  the  maximum  displacement  immediately  about  the 

normal. 

The    maximum    of   the    maxima   displacements  will    occur  when,  in 

Equation  (640), 

2  77  .  6  .  cos  6 
sm  . r = 

A 

and  which  would  reduce  that  equation  to 

c  X 


±i; 


{^P)u  = 


TT  .  /  .  cos  I 


and  as  the  living  forces  are  proportional  to  the  squares  of  the  greatest 
displacements,  we  have 

m  . v;  :  m  .  v.;  :  :  —^k-  :  3— r^ ^a • 

'  "  V  TV  .1    COS-'  d 


40-i  ELEMENTS     0  i'     ANALYTICAL    .MECHANICS. 

Whence 

a' 
m  .V  /  =  7n  .vK- — ^—, — r (G42) 

"  '     in'  b-  .  cos^  0  ^ 

in  whicli  t\  is  the  velocity  of  the  nioleciile  on  the  normal,  and  i\^ 
that  at  the  angular  distance  0  from  it.  When  the  waves  ai'C  \ery 
short,  as  compared  Avith  b,  it  is  obvious  tliat  the  living  force  of  the 
molecules  would  be  sensibly  nothing,  except  immediately  about  the 
normal.  When  the  waves  are  long,  as  compared  wiih  b,  the  li\ing 
force  w'ill  be  appreciable  hr  every  value  of  0,  and,  therefore,  in  every 
direction  in  front  of  the  primitive  wave.  The  importance  of  this 
discussioi    will  1;  c  apparent  in  the  subjects  of  sound  and  light. 


r/ 


PAKT    IV. 


APPLICATION   OF   THE    PRECEDING    PRINCIPLES    TO 
SIMPLE   MACHINES,   PUMPS,   ETC. 

§  349. — Any  device  by  Avhich  the  action  of  ji  force  may  be  received 
at  one  place  and  transmitted  to  another  is  called  a  Machine. 

There  are  usually  seven  elementary  machines  discussed  in  Me- 
chanics;  viz.,  the  Cord,  Lever,  Inclined  Plane,  Pulley,  Screw,  Wheel  and 
Axle,  and  Wedge.  The  Cord,  Lever,  and  Inclined  Plane  are  called 
Simple  Machines  ;  the  others,  being  combinations  of  these,  are  called 
Compound  Machines. 

§  350. — In  Machines,  as  in  all  other  bodies,  every  action  is  ac- 
companied by  an  equal  and  contrary  reaction.  A  force  which  acts 
npon  a  Machine  to  impress  or  preserve  motion  is  called  a  Poa'cr. 
A  force  which  reacts  to  prevent  or  destroy  motion,  is  called  a 
Resistance.  The  Agent  which  is  the  source  of  power,  is,  §38,  called 
a  Motor. 

§  351. — Kesuming  Equation  (30),  and  supposing  the  displacement, 
which  in  that  equation  was  wholly  arbitrary,  to  conform  in  e\(."ry 
respect  to  that  caused  by  the  powers  and  resistances,  we  shall  liave 
6  s  =■  d  s,  s  being  the  path  described  by  the  elementary  mass  m; 
and  hence, 

d^  s 
^  dt"  ' 


but 


whence, 


rf2  s  d  s     d'^s  ,  ,    ,  ,  2> 

— —  ds  =  -—  '  — —  =  vdv  =  id  (v^) ; 
dt^  dt      dt  i      ~    ' 


:E  Pop  -   '^Im.d(r')=-.0. (043) 


406  ELEMENTS     OF     ANALYTICAL    MECHANICS. 

Denoting   by   Q,    Q\  &c.  the  resistances,  by  P,  P'.  &c.    the    pow 
ers,  8  q,  &c.  and    (J^;,  &c.  the   projections    of  their    respective    virtual 
velocities  ;     the    first    term,    v^'hich    embraces    all     the    forces    except 
inertia  in  action  on  the  machine,  may  be  replaced   by  IP  op  —  1  Qoq^ 
and  we   have 

^Pop  —  'S.QSq  —  ^S.m.dv'^.    •     •     •     •     (644) 
Integrating, 

f^PSp  -  J':E  QSq  =  ^^niv^  +   C; 

and   denoting   by  i\  the   initial   velocity,    and   taking   the   integral   sa 
as   to   vanish   when    t  =  0, 

fl.  P  dp  —  J-L  Q8q  zzi^-Zmv"^  —  I'S.m  v,2.    .  .  .  (645) 

The  products  P  dp)  and '  Q  S  q  are  the  elementary  quantities  of 
work  performed  by  a  power  and  a  resistance  respectively,  in 
the  element  of  time  d  t ;  the  product  ^mdv^  is  the  elementary 
quantity  of  work  performed  by  the  inertia,  or  one  half  the  incre 
ment  of  living  force  of  the  mass  7?^  in  this  time.  And  Equation 
(645)  shows  that  in  any  machine,  in  motion,  the  increment  of  the 
half  sum  of  the  living  forces  of  all  its  parts  is  always  equal  tc 
the  excess  of  the  work  of  the  powers  or  motors  over  that  of  the 
resistances. 

§352. — If  the   machine    start   from   rest,  Equation  (645)  becomes 

fEPSp—f:EQSq^^:Smv^,'     •     •     .      (640) 

and  as  the  second  member  is  essentially  positive,  the  work  of  the 
motors  must  exceed  that  of  the  resistances  embraced  in  the  term 
I H  Qo  q ;  in  other  words,  the  inertia  will  oppose  the  motor  and 
act  as  a  resistance.  When  the  motion  becomes  uniform,  the  second 
member  will  be  constant ;  from  that  instant  inertia  will  cease  to 
act,  and  the  subsequent  work  of  the  motor  will  be  equal  to  that 
of  the  resistances  as  long  as  this  motion  continues.  If  the  motion 
be  now  retarded,  the  second  member  will  decrease,  the  inertia  will 
act  with   the   power,  and    this    will    continue   till    the   machine   coni*:s 


APPLICATIOXS.  407 

to  rest,  and  the  excess  <  f  work  of  the  Resistance  during  retardation 
will  be  exactly  equal  to  that  of  the  Poicer  during  acceleration. 
(_Generall^,  then,  when  a  machine  is  at  rest  or  is  moving  unifurinly, 
inertia  does  not  act ;  when  the  motion  is  variable,  it  does,  and 
opposes  or  aids  the  motor  according  as  the  motion  is  accelerated 
or  retarded. 

§353. — The  essential  parts  of  every  machine  are  those  which 
receive  c'rectly  the  action  of  the  motor,  those  which  act  directly 
upon  the  body  to  be  moved  or  transformed,  and  those  which  serve 
to  transmit  the  action.  The  arrangement  of  the  latter  is  often  a 
source  of  resistance,  arising  from  Friction,  Adhesion,  Stiffness  of 
Cordage,    &c.,    whose    work    enters    largely    into    the    general    term 


I 


§  354. — When  two  bodies  are  pressed  together,  experience  shows 
that  a  certain  effort  is  always  required  to  cause  one  to  roll  or  slide 
along  the  other.  This  arises  almost  entirely  from  the  inequalities  in 
the  surfaces  of  contact  interlocking  with  each  other,  thus  rendering 
it  necessary,  when  motion  takes  place,  either  to  break  them  off,  com- 
press them,  or  force  the  bodies  to  separate  far  enough  to  allow  them 
to  pass  each  other.  This  cause  of  resistance  to  motion  is  called  fric- 
tion, of  which  we  distinguish  two  kinds,  according  as  it  accompanies 
a  sliding  or  rolling  motion.  The  first  is  denominated  sliding,  and 
the  second  rolling  friction.  They  are  governed  by  the  same  laws; 
the  former  is  much  greater  in  amount  than  the  latter  under  given 
circumstances,  and  being  of  more  importance  in  machines,  will  prin- 
cipally occupy   our  attention. 

The  intensity  of  friction,  in  any  given  case,  is  measured  by  the 
force  exerted  in  the  direction  of  the  surfece  of  contact,  which  will 
place  the  bodies  in  a  condition  to  resist,  during  a  change  of  state, 
in  respect  to  motion  or  rest,  only  by  their  inertia. 

§355. — The  friction  between  two  bodies  maybe  measured  directly 
by  means  of  the  spring  balance.      For  this   purpose,  let  the  surface 


408 


ELEMENTS  OF  ANALYTICAL  MECHANICS. 


Ci>  of  one   of  the    bodies  M  be   made   perfectly  /evel,  so    that    the 

other  body  M\  when  laid 

upon    it,   may  press    with 

its  entire  weight.  To  some 

point,  as  E,  of  the   body 

M',  attach  a  cord  with  a 

spring     balance     in     the 

manner  indicated  in  the  figure,  and    apply  to   the  latter  a  force  F  of 

such  intensity  as  to  produce  in  the  body  M'  a  uniform  motion.      The 

motion    being    uniform,  the  accelerating  and  retarding  forces  must  be 

equal    and    contrary ;    that  is  to  say,  the  friction    must  be  equal  and 

contrary  to    the  force  F^  of  which    the    intensity  is    indicated    by  the 

balance. 

The  experiments  on  friction  which  seem  most  entitled  to  confi- 
dence are  those  performed  at  Metz  by  M.  Morin,  under  the  orders 
of  the  French  government,  in  the  years  1S31,  1832,  and  1833.  They 
were  made  by  the  aid  of  a  contrivance,  first  suggested  by  M.  Pon- 
celet,  which  is  one  of  the  most  beautiful  and  valuable  contributions 
that  theory  has  ever  made  to  practical  mechanics.  Its  details  are 
given  in  a  work  by  M.  Morin,  entitled  '■'■  Nouvelles  Fxjyeriences  sur  le 
Frottementy     Paris,  1833. 

The  following  conclusions  have  been  drawn  from  these  experi- 
ments, viz. : 

The  friction  of  two  surflices  which  have  been  for  a  considerable 
time  in  contact  and  at  rest  is  not  only  different  in  amount,  but  also 
in  nature,  from  the  friction  of  surfaces  in  continuous  motion  ;  espe- 
cially in  this,  that  the  friction  of  quiescence  is  subjected  to  causes  of 
variation  and  uncertainty  from  which  the  fj'iction  during  motion  is 
exempt.  This  variation  does  not  appear  to  depend  upon  the  extent 
of  the  surface  of  contact;  for,  with  difierent  pressures,  the  ratio  of 
the  friction  to  the  pressure  varied  greatly,  although  the  surfaces  of 
contact  were  the  same. 

The  slightest  jar  or  shock,  producing  the  most  imperceptible 
movement  of  the  surfaces  of  contact,  causes  the  friction  of  quies- 
cence to  pass  to  that  -which  accompanies  motion.  As  every  machine 
may  be  regarded  as  being  subject  to  slight  shocks,  producing  imper 


APPLICATIONS.  409 

ceptible  motions  in  the  ;  urfiices  of  contact,  the  kind  o:'  friction  to  be 
employed  in  all  questions  of  equilibrium,  as  well  as  of  motions  of 
machines,  should  obviously  be  this  la;t  mentioned,  or  that  which 
accompanies  continuous  motion. 

The  LAWS  of  friction  which  accompanies  continuous  motion  are 
remarliably  uniform  and  dejiiiite.     These  laws  are : 

1st.  Friction  accompanying  continuous  motion  of  two  surfaces, 
between  which  no  unguent  is  interposed,  bears  a  constant  proportion 
to  the  force  by  which  those  surflices  are  pressed  together,  whatever 
be  the  intensity  of  the  force. 

2d.  Friction  is  wholly  independent  of  the  extciit  of  the  surfaces  in 
contact. 

3d.  Where  unguents  are  interposed,  a  distinction  is  to  be  made 
between  the  case  in  which  the  surfaces  are  simply  unctuous  and  in 
intimate  contact  with  each  other,  and  that  in  vrhich  the  surfaces  are 
wholly  sejoaraied  from  one  another  by  an  interposed  stratum  of  the 
unguent.  The  friction  in  these  two  cases  is  not  the  same  in  amount 
under  the  same  pressure,  although  the  law  of  the  independence  of 
extent  of  surface  obtains  in  each.  When  the  pressure  is  increased 
sufficiently  to  press  out  the  unguent  so  as  to  bring  the  unctuous  sur- 
faces in  contact,  the  latter  of  these  cases  passes  into  the  fii\st;  and 
this  fact  may  give  rise  to  an  apparent  exception  to  the  law  of  the 
independence  of  the  extent  of  surface,  since  a  diminution  of  the  sur- 
face of  contact  may  so  concentrate  a  given  pressure  as  to  remove  the 
unguent  from  between  the  surfaces.  The  exception  is,  however,  but 
apparent,  and  occurs  at  the  passage  from  one  of  the  cases  above- 
named  to  the  other.  To  this  extent,  the  law  of  independence  of  the 
extent  of  surface  is,  therefore,  to  be  received  with  restriction. 

There  are,  then,  three  conditions  in  respect  to  friction,  under 
which  the  surfaces  of  bodies  in  contact  may  be  considered  to  exist, 
viz.:  1st,  that  in  which  no  unguent  is  present;  2d,  that  in  which 
the  surfaces-  are  simply  unctuous;  3d,  that  in  which  there  is  an 
interposed  stratum  of  the  unguent.  Throughout  each  of  these  states 
the  friction  which  accompanies  motion  is  always  proportional  to  the 
pressure,  but  for  the  same  pressure  in  each,  very  different  in 
amount. 


410 


ELEMENTS     OF    ANALYTICAL    MECHANICS 


4th.  The  friction  which  accompanies  motion  is  always  independ 
ent  of  the  velocity  with  which  the  bodies  move ;  and  this,  whether 
the  surfaces  be  without  unguents  or  lubricated  with  water,  oils, 
grease,  glutinous  liquids,   syrups,  pitch,   &c.,   &c. 

The  variety  of  the  circumstances  under  which  these  laws  obtain, 
and  the  accuracy  with  which  the  phenomena  of  motion  accord  with 
them,  may  be  inferred  from  a  single  example  taken  from  the  first 
set  of  Morin's  experiments  upon  the  friction  of  surfaces  of  oak, 
whose  fibres  were  parallel  to  the  direction  of  the  motion.  The  sur- 
faces of  contact  were  made  to  vary  in  extent  from  1  to  84 ;  the 
forces  which  pressed  them  together  fi'om  88  to  2205  pounds ;  and 
the  velocities  from  the  slowest  perceptible  motion  to  9,8  feet  a 
second,  causing  them  to  be  at  one  time  accelerated,  at  another 
uniform,  and  at  another  retarded ;  yet,  throughout  all  this  wide 
rano-e  of  variation,  in  no  instance  did  the  ratio  of  the  friction  to 
the  pressure  differ  from  its  mean  vahie  of  0,478  by  more  Thiui  gy 
of  this  same  fraction. 

Denote  the  constant  ratio  of  the  entire  friction  F,  to  the  noniud 
pressure  P,  by/;  then  will  the  first  law  of  friction  be  expressed  by 
the  follov,-ing  equation, 

|-=/; (c^') 

whence, 

F=f.P. 

This  constant  ratio  /  is  called  the  co-efficient  of  friction.,  because, 
when  multiplied  by  the  total  normal  pressure,  the  product  gives 
the    entire  friction. 

Assuming  the  first  law  of  fric- 
tion, the  co-efficient  of  friction  n:iay 
easily  be  obtained  by  means  of  the 
inclined  plane.  Let  W  denote  the 
weight  of  any  body  placed  upon 
the  inclined  plane  A  B.  Eesolve 
this  weight  G  G'  into  two  compo- 
nents, one  GM  perpendicular  to 
the   plane,  and   the  other   G  N  par- 


APPLICATIONS.  411 

allel    to   it.      Because  the   angles    G' G M  and  BAC  are   equal,  the 
first   of  these   comporents  will  be 

GM  =  PT.cos^, 
and   the   second, 

GN=  W.smA, 
in  which  A   denotes   the   angle  BA  C. 

The  first  of  these  components  determines  the  total  pressure  upon 
the   plane,  and   the   friction  due   to    this   pressure  will   be 

F  z^f.W  cos  A. 

The  second  component  urges  the  body  to  move  down  the  plane. 
If  the  inclination  of  the  plane  be  gradually  increased  till  the  body 
move  with  uniform  motion,  the  total  friction  and  this  component 
must   be    equal    and    opposed ;    hence, 

/.  TF.  cos  ^  —  W .  sin  A  ; 

whence, 

.       sin  yl 

/  =  ;  =  tan  A. 

cos  ^1 

We,  therefore,  conclude,  that  the  unit  or  co-efficient  of  friction 
between  any  two  surfiices,  is  equal  to  the  tangent  of  the  angle 
which  one  of  the  surfaces  must  make  with  the  horizon  in  ordei 
that  the  other  may  slide  over  it  with  a  uniform  motion,  the  body 
to  which  the  moving  surface  belongs  being  acted  upon  by  its  own 
weight  alone.  This  angle  is  called  the  angle  of  friction  or  limiting 
angle    of  resistance. 

The  values  of  the  imit  of  friction  and  of  the  limiting  angles  for 
many  of  the  various  substances  employed  in  the  art  of  construction, 
are  given  in  Tables  VI,  VII    and  VIII. 

The  distinction  between  the  friction  of  surfaces  to  which  no  un 
guent  is  applied,  those  which  are  merely  unctuous,  and  those  between 
which  a  uniform  stratum  of  the  unguent  is  interposed,  appears  first 
to  have  been  remarked  by  M.  Morin ;  it  has  suggested  to  him 
what  appears  to  be  the  true  explanation  of  the  difference  between 
his    results    and    those    of  Coulomb.      He   conceives,  that   in    the    ex- 


4:12  ELEMENTS     OF     ANALYTICAL     MECHANICS. 

periments  of  this  celebrated  Engineer,  the  requisite  precautions  had 
not  been  taken  to  exclude  unguents  from  the  surfaces  of  contact. 
The  slightest  unctuosity,  such  as  might  present  itself  accidentally, 
vmless  expressly  guarded  against — such,  for  instance,  as  might  have 
been  left  by  the  nands  of  the  workman  who  had  given  the  last 
polish  to  the  surfaces  of  contact — is  sufiicient  materially  to  affect 
the    co-efficient  of  friction. 

Thus,  for  instance,  surfaces  of  oak  having  been  rubbed  Avith  hard 
dry  soap,  and  then  thoroughly  wiped,  so  as  to  show  no  traces 
whatever  of  the  unguent,  were  found  by  its  presence  to  have  lost 
l"*^*  of  their  friction,  the  co-efficient  having  passed  from  0,478 
to  0,164. 

This  effect  of  the  unguent  upon  the  friction  of  the  surfaces  may 
be  traced  to  the  fact,  that  their  motion  u])on  one  another  without 
unguents  was  always  found  to  be  attended  by  a  wearing  of  both  the 
surffices  ;  small  particles  of  a  dark  color  continually  separated  from 
ihem,  Avhich  it  was  found  from  time  to  time  necessary  to  remove, 
md  which  manifestly  influenced  the  friction  :  now,  with  the  presence 
of  an  unguent  the  formation  of  these  particles,  and  the  consequent 
wear  of  the  surfaces,  completely  ceased.  Instead  of  a  new  surface 
of  contact  being  continually  presented  by  the  wear,  the  same  surface 
remained,  receiving    by  the  motion  continually  a  more  perfect  polish. 

A  comparison  of  the  results  enumerated  in  Table  VIII,  leads  to 
the  following  remarkable  conclusion,  easily  fixing  itself  in  the  memory, 
that  loith  the  nntjuents,  hogs'  lard  and  olive  oil  -inierjiosed  in  a  con- 
iinitoNS  stratum  between  them,  surfaces  of  toood  on  metal,  loood  on 
wood,  metal  on  wood,  and  metal  on  metal,  token  in  motion,  have  all 
of  them  verij  nearly  the  same  co-cfficient  of  friction,  the  value  of  that 
co-efficient  being  in  all  cases  included  between  0,07  and  0,08,  and  the 
limiting   angle    of  resistance   therefore  between  4°  and  4°  35'. 

For  the  unguent  talloiu  the  co-efficient  is  the  same  as  the  above  in 
every  case,  except  in  that  of  metals  zq^on  metals ;  this  unguent  seems 
less  suited  to  metallic  surfaces  than  the  others,  and  gives  for  the 
mean  value  of  its  co-efficient  0,10,  and  for  its  Vmiiting  angle  of  re- 
sistance 5°  43'. 


APPLICATTOXS. 


413 


S56. — Besides  friction,  there  is  another  cause  of  resistance  to  the 
motion  of  bodies  when  moving  over  one  another.  The  same  forces 
which  hold  the  elements  of  bodies  together,  also  tend  to  keep  the 
bodies  themselves  together,  when  brought  into  sensible  contact.  The 
elibrt  by  which  two  bodies  are  thus  united,  is  called  the  force  of 
Adhesion. 

Familiar  illustrations  of  the  existence  of  this  force  are  furnished 
by  the  pertinacity  with  which  sealing-wax,  wafers,  ink,  chalk  and 
black-lead  cleave  to  paper,  dust  to  articles  of  dress,  paint  to  tiie 
surface    of  wood,  whitewash    to    the   walls    of  buildings,  and   tlic  like. 

The  intensity  of  this  force,  arising  as  it  does  from  the  afliiiity 
of  the  elements  of  matter  for  each  other,  must  vary  with  the  num- 
ber of  attracting  elements,  and  therefore  with  the  exient  of  the  mir- 
Jace  of  contact. 

This  law  is  best  verified,  and  the  actual  amount  of  adhesion  be- 
tween different  substances  determined,  by  means 
of  a  delicate  spring-balance.  For  this  purpose, 
the  surflices  of  solids  are  reduced  to  polished 
planes,  and  pressed  together  to  exclude  the  air, 
and  the  efforts  necessary  to  separate  them  noted 
by  means  of  this  instrument.  The  experiment 
being  often  repeated  with  the  same  substances, 
laving  different  extent  of  surfaces  in  contact,  it 
is  found  that  the  effort  necessary  to  produce 
the  separation  divided  by  the  area  of  the  surface 
gives  a  constant  ratio.  Thus,  let  S  denote  the 
area  of  the  surfaces  of  contact  expressed  in  square 
feet,  square  inches,  or  any  other  superficial  unit; 
A  the  eflbrt  required  to  separate  them,  and  a 
the    constant   ratio  in   question,  then  will 

A 

-s  =  '^ 


or. 


A  =  a.S. 
The    constant   a   is   called   the  unit   or  co-efficient  of  adhesion,  and  ob- 


il4 


ELEMENTS     OF    ANALYTICAL    MECHANICS. 


viously  expresses  the  value  of  adhesion  on  each  unit  of  surface,  for 
making 

we   have 

A  =  a. 

To  find  the  adhesion  between  solids  and  liquids,  suspend  the  solid 
from  the  balance,  with  its  polished  surfece  downward  and  in  a  hori- 
zontal position  ;  note  the  weight  of  the  solid, 
then  bring  it  in  contact  with  the  horizontal 
surface  of  the  fluid  and  note  the  indication  of 
the  balance  when  the  separation  takes  place, 
on  drawing  the  balance  up  ;  the  difference  be- 
tween this  indication  and  that  of  the  weight 
will  give  the  adhesion ;  and  this  divided  by 
the  extent  of  surface,  will  give,  as  before,  the 
cu-cfRcient  a.  But  in  this  experiment  two 
opposite  conditions  must  be  carefully  noted, 
else  the  cohesion  of  the  elemer.ts  of  the  liquid 
for  each  other  may  be  mistaken  for  the  adhe- 
sion of  the  solid  for  the  fluid.  If  the  solid 
on  being  removed  take  with  it  a  layer  of  the 
fluid ;    in    other    words,    if    the    solid    has   been 

wet  by  the  fluid,  then  the  attraction  of  the  elements  of  the  solid 
for  those  of  the  liquid  is  stronger  than  that  of  the  elements  of  the 
liquid  for  each  other,  and  a  will  be  the  unit  of  adhesion  of  two 
surfaces  of  the  fluid.  If,  on  the  contrary,  the  solid  on  leaving  the 
fluid  be  perfectly  dry,  the  elements  of  the  fluid  will  attract  each 
other  more  powerfully  than  they  will  those  of  the  solid,  and  a  will 
denote   the   unit   of  adhesion  of  the   solid   for   the   liquid. 

It  is  easy  to  multiply  instances  of  this  diversity  in  the  action  of 
solids  and  fluids  upon  each  other.  A  drop  of  water  or  spirits  of 
wine,  placed  upon  a  wooden  table  or  piece  of  glass,  loses  its  globu- 
lar form  and  spreads  itself  over  the  surfoce  of  the  solid  y  a  drop  of 
mercury  will  not  do  so.  Immerse  the  finger  in  water,  it  becomes 
wet ;    in    quicksilver,  it   remains   dry.     A   tallow  candle,  or  a  feather 


APPLICATIONS. 


4:15 


from  anj  species  of  water-fowl,  remains  dry  the  jgli  dipped  in  water. 
Gold,  silver,  tin,  lead,  &c.,  become  moist  on  being  immersed  in 
quicksilver,  but  iron  and  platinum  do  not.  Quicksilver  when  poured 
into  a  gauze  bag  will  not  run  through ;  water  will :  place  the  gauze 
containing  the  quicksilver  in  contact  with  water,  and  the  metal  will 
also  flow  through. 

It  is  difficult  to  ascertain  the  precise  value  of  the  force  of  adhe 
sion  between  the  rubbing  surfaces  of  machinery,  apart  from  that  of 
friction.  But  this  is  attended  with  little  practical  inconvenience,  as 
long  as  a  machine  is  in  motion.  The  experiments  of  wliicli  the 
results  are  given  in  Tables  VI,  VII  and  VIII,  and  which  are  applicable 
to  machinery,  were  made  under  considerable  pressures,  such  as  those 
with  vrhich  the  jDarts  of  the  larger  machines  are  accustomed  to  move 
upon  one  another.  Under  such  pressures,  the  adhesion  of  unguents 
to  the  surfaces  of  contact,  and  the  opposition  to  motion  presented 
by  their  viscosity,  are  causes  whose  influence  may  be  safely  disre 
garded  as  compared  with  that  of  friction.  In  the  cases  of  lighter 
machinery,  however,  such  as  watches,  clocks,  and  the  like,  these 
considerations  rise   into    importance,  and    cannot    be    neglected. 


J-^ 


STIFFNESS    OF    CORDAGE. 


§  357. — Conceive  a  wheel  turning 
freely  about  an  axle  or  trunnion,  and 
haviuo;  in  its  circumference  a  groove  to 
receive  a  cord  or  rope.  A  weight  IF, 
being  suspended  from  one  end  of  the 
rope,  while  a  force  F,  is  applied  to  the 
other  extremity  tu  draw  it  up,  t!ie 
latter  will  experience  a  resistance  in 
consequence  of  the  rigidity  of  the  rope, 
which  opposes  every  effin-t  to  bend  it 
around  the  wheel.  This  resistance  must, 
of  necessity,  consume  a  portion  of  the 
W(jrk  of  the  force  F.  The  measure  of 
the    resistance    due    to    the    rigidity  of   cordage    has    been    made    the 


416  ELEMENTS     OF     ANALYTICAL    MECHANICS. 

subject  of  experiment  by  Coulomb ;  and,  according  to  him,  it 
results  that  for  the  same  cord  and  same  wheel,  this  nieasnre  is 
composed  of  two  parts,  of  which  one  remains  constant,  while  the 
other  varies  with  the  weight  IF,  and  is  directly  proportional  to  it; 
so  that,  designating  the  constant  part  by  /i,  and  the  ratio  of  the 
vari{|j^e  part  to  the  weight  W  by  /,  the  measure  will  be  given  by 
the  expression 

in  which  Jt  represents  the  stiffness  arising  from  the  natural  torsion 
or  tension  of  the  threads,  and  /  the  stiffness  of  the  same  cord  due  to 
a  tension  resulting  from  one  unit  of  weight ;  for,  making  W  =  1,  the 
above  becomes 

X  +  /. 

Coulomb  also  foxmd  that  on  chai]ging  the  wheel,  the  stiffness  varied 
in  the  invei'se  ratio  of  its  diameter ;   so  that  if 

be  the  measure  of  the  stillness  for  a  wheel  of  one  foot  diameter,  then 

will 

K  -\-  I.  W 

be  the  measure  when  the  wheel  has  a  diameter  of  2  E.  A  table 
giving  the  values  of  IC  and  /  for  all  ropes  and  cords  employed  in 
practice,  when  wound  around  a  -wheel  of  one  foot  diameter,  and  sub- 
jected to  a  tension  arising  from  a  uin't  of  weight,  would,  therefore, 
enable  us  to  find  the  stiffness  answering  to  any  other  wheel  and 
weight  whatever. 

But  as  it  Avould  be  impossible  to  anticipate  all  the  ditTerent  sizes 
of  ropes  used  under  the  various  circumstances  of  practice,  Coulomb 
also  ascertained  the  law  which  connects  the  stiffness  with  the  diame- 
ter of  the  ci'oss-section  of  the  rope.  To  express  this  law  in  all  cases, 
he  found  it  necessary  to  distinguish,  1st,  new  ivhite  rope,  either  dry 
or  moist ;  2d,  white  ropes  partly  worn,  either  dry  or  moist ;  od,  tarred 
ropes  ;  4th.  packthread.  The  stiffness  of  the  first  class  he  found  nearly 
proportional  to  the  square  of  the  diameter  of  the  cross-section  ;    that 


APPLICATIONS.  417 

of  the  second,  to  the  square  root  of  the  cube  of  this  diameter,  near]v; 
that  of  the  third,  to  the  number  of  yarns  in  the  rope ;  and  that  of 
the  fourth,  to  the  diameter  of  the  cross-section  So  that,  if  S  denote 
the  resistance  due  to  the  stiffness  of  any  given  rope;  d  ihe  ratio  of 
its  diameter  to  that  of  the  table;  and  n  the  ratio  of  the  nunibei  of 
yarns  in  anv  tarred  rope  to  that  of  the  table,    we  shall  have  tor 

jVeic  white  rope,  dry  or  moist. 

^  =  ''-'^-- •    •    ("^^) 

Half  worn  white  rope,  dry  or  moist. 

S^d'--         2A>    ~ (^^^) 

Tarred  rope. 

K  +  I  .W  ,       , 

^=''"-\e — C^^^^ 

Packthread. 

^    K-\-  I.W  ,      ^ 

For  packthread,  it  will  always  be  sufficient  to  use  the  tabular 
values  given,  corresponding  to  the  least  tabular  diameters,  and  substi- 
tute them  in  Equation  (651).  An  example  or  two  will  be  sufficient 
to  illustrate  the  use  of  these  tables. 

U.vainple  Isi.  Required  the  resistance  due  to  the  stiffness  of  a  new 
dry  white  rope,  whose  diameter  is  1,18  inches,  when  loaded  with 
a  weight  of  882  j)ounds,  and  wound  about  a  wheel  1,64  feet  in 
diameter. 

Seek  in  No.  1,  Table  X,  the  diameter  nearest  that  of  the  given 
rope  ;    it  is  0.79  ;    hence, 

and  frf.m   the   table  at   the  side, 

d^  =  2,25. 
From  No.   1,  opposite  0,79,  we  find 

X=  1,6097, 

/   =  0,03195; 

27 


418  ELEMENTS     OF    ANALYTICAL    MECHANICS, 

ft- 
which,  together    -svith    the    weight    W  =  882   lbs.,    and   2  B  =  1,64, 

substituted  in  Equation  (648),  give 

S  =  2.25  .  '•""^^  +  ^'"^'^^  X  S^^  =  40^17, 
l,b4 

which  is  the  true  resistance  due  to  the  stiffness  of  the  rope  in 
question. 

Example  2d.  What  is  the  resistance  due  to  the  stiffness  of  a 
white  rope,  half  worn  and  moistened  with  water,  having  a  diam- 
eter equal  to  1.97  inches,  wound  about  a  wheel  0,82  of  a  foot  in 
diameter,  and  loaded  with  a  weight  of  2205  pounds  ? 

The  tabular  diameter  in  No.  4,  Table  X,  next  less  than  1,97, 
is  1,57,  and  hence, 

d  =  ^-^  =  1,3  nearly; 

the  square  root  of  the  cube  of  which  is,  by  the  table  at  the  side, 

d^  =  1,482. 
In  No.  4  we  find,  opposite   1,57, 

IT  =  6,4324, 

/    =  0,06387 ; 

ft. 
which    values,    together    with    W  =  2205   lbs.,   and   2  JR  =  0,82,   in 

Equation  (649),  give 

lbs.  lbs. 

„       ,  ...-,  ^  6,4324  +  0,06387  x  2205  «- 

S  =  1,482  X  ~~0S^ ~  266,109, 

which  is  the  required  resistance. 

Example  od.  What  is  the  resistance  due  to  the  stiffness  of  a 
tarred  rope  of  22  yarns,  when  subjected  to  the  action  of  a  weight 
equal  to  4212  pounds,  and  wound  about  a  wheel  1,3  feet  diameter, 
the  weight  of  one  runnirg  foot  of  the  rope  being  about  0,6  of  a 
pound  % 

By  referring  to  No.  5,  Table  X,  we  find  the  tabular  number  of 
yarns  next  less  than  22  to  be  15,  and  hence, 

22 

«,'=-;;=:  1.466  ncarlv. 
lu  '  -^ 


APPLICATIONS 


419 


In  the  same  table,  opposite  15,  we  find 

K  =  0,7664, 
/   =  0,019879; 

which,  together  with   W  =  4212,  and  2  R  =  1,3,  in  Equation  (650), 
give 

S 


ft- 


1,466  0.^664  +  0,019879  X  4213  ^  ^^^^^ 


1,3 

Example  4th.  Required  the  resistance  due  to  the  stiffness  of  a 
new  white  packthread,  whose  diameter  is  0,196  inches,  when  moist- 
ened or  wet  with  water,  wound  about  a  wheel  0,5  of  a  foot  in 
diameter,   and  loaded  with  a  weight  of  275  pounds. 

The    lowest   tabular    diameter   is  0,39  of  an    inch,  and   hence 

0,196 


d  = 


0,390 


0,5  nearly. 


In    No.  2,  Table    X,  we  find,  opposite  0,39, 

lb. 

K  =z  0,8048, 
/  =  0,00798 ; 

which,  with    W  =  275,  and   2E  =  0,5,  we  find,  after   substituting  in 
Equation  (651), 


S  =  0,5 


0,8048  +  0,00798  x  275 

ois 


=  2,999. 


§  358. — The  resistance  just  found 
is  expressed  in  pounds,  and  is  the 
amount  of  weight  which  would  be 
necessary  to  bend  any  given  rope 
around  a  vertical  wheel,  so  that 
the  portion  A  E,  between  the  first 
point  of  contact  A,  and  the  point 
E,  where  the  rope  is  attached  to 
the  weight,  shall  be  perfectly  straight. 
The  entire  process  of  bending  takes 
place  at  this  first  or  tangential 
point    A  ;    for,  if  motion    be    com- 


420         ELEMEXTS     OF    ANALYTICAL     MECHANICS. 

municated  to  the  wheel  iu  the  direction  indicated  by  the  ai row- 
head,  the  rope,  supposed  not  to  slide,  will,  at  this  point,  taice  and 
retain  the  constant  curvature  of  the  wheel,  till  it  passes  from  the 
latter  on  the  side  of  the  power  F.  When,  therefore,  by  the  motion 
of  the  wheel,  the  point  m  of  the  rope,  now  at  the  tangential  point, 
passes  to  vi\  the  working  point  of  the  force  S  will  have  described 
in  its  own  direction  the  distance  AD.  Denoting  the  arc  de.-;cribed 
bv  a  point  at  the  unit's  distance  from  the  centre  of  the  wheel 
b\    6', ,  and   the   radius   of  the    wheel    by  R.  we    shall    have 

AD  T=  lis^; 

and  representing   the    quantity  of  work  of  the  force  S  by  L.  we  get 

L  =  S.Rs^; 

replacing  S  by  its  value  in  Equations  (648)  to  (651), 

r,  .       K  +    I.    W 

L=^Rs,-d, ^^ (652) 

in  which  d^  represents  the  quantity  f/-,  d^.  n,  or  d,  in  Equations  (648) 
to  (651),  according    to    the   nature  of  the    rope. 

Exam2)le. — Taking  the  2d  example  of  §357,  and  supposing  a  por- 
tion of  the  rope,  equal  to  20  feet  in  length,  to  have  been  brought 
in    contact  with    the    wheel,  after    the   motion    begins,  we  shall  have 

Z  =  20  X  206,109  =  5322,18    units  of  work; 

that  is,  the  cpiantity  of  work  consumed  by  the  resistance  due  to 
the  stiffness  of  the  rope,  while  the  latter  is  moving  over  a  distance 
of  20  feet,  would  be  sufficient  to  raise  a  weight  of  5322.18  pounds 
throuch    a  vertical    height  of  one  foot. 


FniCTION    ox    PIVOTS,    AND   TEUNNICNS. 

§  350. — All  rotating  pieces,  such  as  wheels  supported  upon  other 
pieces,  give  rise  by  their  motion  to  friction.  This  is  an  imjiorlant 
element  in  all  comj)utations  relating  to  the  performance  of  machinery. 
It    seems    to    be    diiferent    according    as    the    rotating   pieces  are    kept 


APPLICATIONS. 


421 


in  pljioe  by  trunnions  or  by 
p.  uots.  By  trunnions  are  meant 
cyruidrical  projections  a  a  from 
the  ends  of  the  ai'bor  A  B  of  a 
vvlieel.  The  trunnions  rest  on  the 
(•oncave  surfaces  of  cylindrical 
jjoxes  CZ>,  with  which  they  usu- 
ally have  a  small  surface  of 
contact  m,  the  linear  elements 
of  both  being  parallel.  Pivots 
are  shaped  lilce  the  trunnions, 
but  support  the  weight  of  the 
wheel  and  its  arbor  upon  their 
circular  end,  which  rests  against 
the  bottom  of  cylindrical  sock- 
ets FGHI. 


Let  iV  denote  the  force,  in  the  direction  of  the  axis,  by  -which 
the  pivot  is  pressed  against  the 
bottom  of  the  socket.  This  force 
may  be  regarded  as  passing 
through  the  centre  of  the  cir- 
cular end  of  the  pivot,  and  as 
the  resultant  of  the  partial  pres- 
sures exerted  upon  all  the  ele- 
mentary surfaces  of  which  this 
circle  is  composed.  Denote  by 
A  the  area  of  the  entire  circle, 
then  will  the  pressure  sustained 
by  ea(;h  unit  of  surface  be 

N 

A  ' 

and    the   pressure  on  any  small  portion  of   the  surface  denoted  by  ci, 

will  obviously  be 

a.  N 


422  ELEMENTS     OF     ANALYTICA'^     MECHANICS. 


and  the  friction  on  the  same  will  be 

f.a.N 
A      ' 

This  friction  may  be  regarded  as  applied  to  the  centre  of  the  ele- 
mentary surface  a;  it  is  opposed  to  the  motion,  and  the  direction  of 
its  action  is  tangent  to  the  circle  described  by  the  centre  of  the 
element.  Denote  the  radius  of  this  circle  by  x,  then  will  the  mo- 
ment of  the  friction  be 

Now,  if  .«  denote  the  length  of  any  variable  portion  of  the  circumfer- 
ence at  the  unit's  distance  from  the  centre   (7,  then  will 


a]  so. 


=.  X  ,  d  s  .  d  X 


which  substituted  above  give 

/.iV. 
and  by  integration. 


dx  . 


'g  .B 


x^  d  X    I        d  s 


Avhence  we  conclude,  that,  in  the  fric- 
tion of  a  pivot,  ice  may  regard  the 
whole  friction  due  to  the  iwessure  as 
acting  in  a  single  2^oint,  and  at  a  dis- 
tance from  the  centre  of  motion  equal 
to  tioo-thirds  of  the  radiiis  of  the  base 
of  the  pivot.  This  distance  is  called 
th'.'  )i/ean  lever  of  friction. 

^  3G0. — If  the  extremity  of  the  pivot, 
instead  of  rubbing  upon  an  entire  circle, 
is  only  in  contact  with  a  ring  or  sur- 
face Comprised    between  t\vo  concentric 


(G53) 


APPJ.ICATIONS.  423 

circles,  as  when   the    arbor    of  a  wheel    is   urged   in    the  direction  of 
its  length  by  the  force  JV  against  a  shoulder  dcha;    then  will 

and  the  integration  will  give 

X"  d  X   I        d  s 
p'              ^„                                   723  ._  7?'3 
f.N-  — -  ^f'  N '  - ---  • 

•^  *    (i22    _    R'-^)  -    3  /  ^2    _    ^/2   ' 

in    which  R    denotes    the    radius    of    the    larger,  and  R'  .that    of  the 
smaller  circle. 

Finally,  denote  by  I  the  breadth  of  the  ring,  that  is,  the  dis- 
tance ^4'  A  ;  by  r,  its  mean  radius  or  distance  from  C  to  a  j)oint 
half  way  between  A'  and  A,  and   we  shall  have 

B    =  r  +  i  /, 

R'  =  r  -  \l; 

substituting   these  values  aljove  and  reducing,  we   have 

/-- 


and  makinii 


+  A 


^:r-7]' (''^) 


we  obtain,  for  the  moment  of  the  friction  on  the  entire  ring, 

f-^-r, (G55) 

The  quantity  r^  is  called  the  mean  lever  of  friction  for  a  ring.  Since 
the  whole  friction  f N  mav  be  considered  as  applied  at  a  point 
whose  distance  from  the  centre  is  f  R.  or  i\  =  r  +  T^""'  according 
as  the  friction  is  exerted  over  an  entire  circle  or  over  a  ring, 
and  shice  the  path  described  by  this  point  lies  always  in  the  di- 
rection in  which  the  tl-iction  acts,  the  quantity  of  work  consumed 
by  it  will  be  equal  tj  the  product  of  its  intensity  fN  into  this 
path.  Designating  the  length  of  the  arc  described  at  the  miit's 
distance  from    C  by  .s-^ ,  the    path    in    (piestion  will   be    either 

§  ^'  •-■; ,     or     r^  s^  ; 


42,1:  ELEMENTS     OF    ANALYTICAL    MECHANICS, 

and    the    quantity  of  work  either 

for   an   entire   circle,   or 


^•^(''  +  127)^' 


for  a  ring.  Let  Q  denote  the  quantity  of  \vork  consumed  by  fric- 
tion in  the  unit  of  time,  and  n  the  number  of  revolutions  jDerformed 
by  the  pivot   in    the   same   time ;    then  will 

and  we   shall    have 

Q  =  ^if  .E.f.Ii.n (650) 

for   the    circle,  and 

^  =  2*-/-i\^-  (r  +  ^)  .«      ....     (057) 

for    a   ring  ;   in  which  <rc  =  3,1416. 

The  co-efficient  of  friction  /,  when  employed  in  either  of  the  fore- 
going   cases,  must   be    taken  from  Table  VI,  VII,  or  VIII. 

Kvam2)Ie. — Eequired  the  moment  of  the  friction  on  a  pivot  of 
cast  iron,  working  into  a  socket  of  brass,  and  which  supports  a 
weight  of  1784  pounds,  the  diameter  of  the  circular  end  of  the 
pivot  being  6  inches.     Here 

in.  ft. 

ig  =  f  =  3  =  0,25, 

lbs. 

iV  =  1784, 
/  =  0,147  ; 

which,  substituted    in  Equation  (653),  gives 

lbs.  ft. 

0,147  X  1784  X  I-  X  0,25  =  43,708. 

And  to  obtain  the  quantity  of  work  in  one  unit  of  time,  say  a 
minute,  there  being  20  revolutions  in  this  unit,  we  make  n  =  20, 
and  *  =  3,1410  in  Equation  (056),  and  find 

Q  =  i  X  3,141G  X  0,25  X  0,147  x  1784  x  20  =^  5402,80; 


APPLICATIOISrs.  42.'i 

that  is  to  say,  during  each  unit  of  time,  there  is  a  quantity  of 
worli  lost  which  would  be  sufficient  to  raise  a  weight  of  5492  80 
pounds  through    a  vertical    distance  of  one  foot. 

Example, — Eequired  the  moment  of  friction,  when  the  pivot  sup- 
ports a  weight  of  2046  pounds,  and  woi'ks  upon  a  shoulder  whose 
exterior  and  interior  diameters  are  respectively  6  and  4  inches ;  the 
pivot   and    socket   being    of  cast    iron,  with  water    interposed. 

L  = —  =  1  mch, 

r  =  2  +  0,.5  ^  2,5  inches, 

(1)2  in.  ft. 

r,  =  2,5  +  J2V25  =  2,5333  =  0,2111, 

JV  =  2046  pounds, 
/=  0,314; 
;i'hich,  substituted  in  Expression  (655),  gives  for  the  moment  of  friction, 

lb.'!.         ft. 

0,314  X  2046  X  0,2111  =  135,62. 

The  Cjuantity  of  work  consumed  in  one  minute,  there  being  sup- 
posed 10  revolutions  in  that  unit,  will  be  found  by  making  in 
Equation  (657),  *  —  3,1416  and  n  =  10, 

Q  =  2  X  3,1416  X  0,314  X  2046  X  0,211  X  10  =  8517,24; 

that  is  to  say,  friction  will,  in  one  unit  of  time,  consume  a  quantity 
of  work  which  would  raise  8517,24  pounds  through  a  vertical  dis- 
tance of  one  foot.  The  quantity  of  work  consumed  in  any  given 
time  would  result  from  multiplying  the  work  above  found,  by  the 
time    reduced   to   minutes. 

TRUNNIONS. 

§  361. — The  friction  on  trunnions  and  axles,  which  we  now  pro- 
ceed to  consider,  gives  a  considerably  less  co-efficient  than  that  which 
accompanies  the  kinds  of  motion  referred  to  in  §  355.  This  will 
appear  from  Table  IX,  which  is    the    result  of  careful    experiment. 

The  contact  of   the    trunnion    with    its    box    is    along    a    linen r   ele- 


i26 


ELEMENTS     OF    ANALYTICAL    MECHANICS. 


ment,  common  to  the  surftices  of  both.  A  section  perpendicular  to 
its  length  would  cut  from  the  trunnion  and  its  box,  two  circles  tan- 
gent  to  each  other  internally.  The  trunnion  being  acted  on  only  by 
its  weight,  would,  when  at  rest,  give  this  tangential  point  at  o,  the 
lowest  point  of  the  section  p  o  g  of  the  box.  If  the  trunnion  be  put 
in  motion  by  the  application  of  a  force,  it  would  turn  around  the 
point  of  contact  and  roll 
indefinitely  along  the  sur- 
face of  the  box,  if  the 
latter  were  level ;  but  this 
not  being  the  case,  it  will 
ascend  along  the  inclined 
surface  o^  to  some  point 
as  m,  where  the  inclina- 
tion of  the  tangent  u  m  v 
is  such,  that  the  friction 
is  just  sufficient  to  pre- 
vent the  trunnion  from  sliding.  Here  let  the  trunnion  be  in  equili- 
brio.  But  the  equilibrium  requires  that  the  resultant  of  all  the 
forces  which  act,  friction  included,  shall  pass  through  the  point  m 
and  be  normal  to  the  surface  of  the  trunnion  at  that  point.  The 
friction  is  applied  at  the  point  m ;  hence  the  resultant  N  of  all  the 
other  forces  must  pass  through  m  in  some  direction  as  md\  the 
friction  acts  in  the  direction  of  the  tangent;  and  hence,  in  order 
that  the  resultant  of  the  friction  and  the  force  N  shall  be  normal  to 
the  surface,  the  tangential  component  of  the  latter  must,  when  the 
other  component  is  normal,  be  equal  and  directly  opposed  to  the 
friction. 

Take  upon  the  direction  of  the  force  N  the  distance  m  d  to 
represent  its  intensity,  and  form  the  rectangle  a  d  b  m,  of  which 
the  side  m  b  shall  coincide  with  the  tangent,  then,  denoting  the 
angle  dm  a  by  9,  will  the  component  of  N  perpendicular  to  the  tan- 
gent be 

iV  .  cos  9  ; 

and  the  friction  due  to  this  pressure  will  be 

f .  N .  cos  9. 


APPLICATIONS. 


+2: 


(658) 


The  component  of  N,  in  the  direction  of  the  tangent,  will  be 

iV .  sin  cp  ; 
and  as  this  must  be  equal  to  the  friction,  we  have 

f .  N .  cos  cp  —  N .  sin  9  j 

whence, 

/  =  tan  (p  ; 

that  is  to  say,  tlte   ratio   of   the  /fiction    to    the  pressure   on    the    trun- 
nion   is  equal    to    the    tangent  of  the    angle    which    the   direction    of  the 
resultant  iY,  of  all  the  forces    except    tlie  friction^   makes    'witli    the    nor- 
mal   to    the    surface    of    the    trunnion    at 
the  point  of  contact.     This  gives  an  easy 
method    of    fmding     the    point    of    con- 
tact.      For   this   purpose,    we    have   bat 
to    draw    through    the   centre    A   a    line 
A  Z,    parallel    to     the    direction    of    iV, 
and    through   A   the    line    Am^  making 
with  A  Z   an   angle   of  which    the    tan- 
gent  is  /;    the  point   «?,   in   which   this 
line     cut's     the    circular    section   of   the 
trumiion,  will   be  the  point  of  contact. 

Because  m-  a  d  b,  last  figure,  is  a  rectangle,  we  have 

iV2  —  m  CO32  (p  ^_  jV2  sin2  (p  ; 
and,   substituting  for  iV- sin^  9   its  equal  /^  iY2  cos^  (p^   we  have 

iV2    =    iV^COS^^    +/2jV2cos2(p    =    jV2cos2  9   (1    +  /^)  ; 


whence. 


iV^cos  (p  =  N  X 


1 


and  multiplying  both   members  by  /, 
f .  JV  .  cos  (p  =  ^  • 


(650) 


-/T  +  /2 

but    the    first   member    is    the     total    friction  ;    whence    we     conclude 
that   to  find   tJie  friction   rf^yon   a  trunnion^  we  have  but  to  mnltiphj  ihe 


428 


ELEMENTS     OF     ANALYTICAL     MECHANICS. 


resultant  of  the  forces  ichlck  act  xvpon  it  by  the  unit  (f  friction^  found 
in  Table  JX,  and  divide  this  product  by  the  square  root  of  tlie  square 
of  this   same  unit  increased  by  unity. 

This  friction   acting  at  the  extremity  of  the  radius  H  of  the   trun- 
nion  and  in  the  direction  of  the  tangent,   its  moment  will   be 


iV 


/ 


V^+f 


X  n. 


(660) 


And   the  path  described  by   the    point  of   application  of    the    friction 
being  denoted   by  Hs^,  the  quantity  of  work  of  the  friction  will  be 


JV.  F.  .s,  X 


/ 


VTT/-' 


(6G1) 


in  which  s^  denotes  the  2:)ath  described  by  a  point  at  the  unit's  dis- 
tance from  the  centre  of  the  truimion.  Denoting,  as  in  the  case  of 
the  pivot,  the  number  of  revolutions  performed  by  the  trunnion  in 
a  unit  of  time,  say  a  minute,  by  71 ;  the  quantity  of  w'ork  perfoi-med 
by  friction  in  this   time  by   Q^;    and  making  -tt  =  3,1416,  we  have 


and 


2ir .  n\ 


Q^  ^^ir  .B.n.N. 


f 


(602) 


When   the   trunnion   remains   fixed   and   does    not   form   part   of    the 
rotating   body,   the   latter  will    turn    about    the    trunnion,    wdiich    now 
becomes    an    axle,    having    the   centre    of 
motion  at  A,   the    centre    of    the    eye   of 
the  wheel ;  in  this  case,  the  lever  of  fric- 
tion   becomes    the   radius    of    the    eye    of 
the     wheel.       As    the    quantity    of    woi'k 
consumed    by    friction     is     the     greater. 
Equation     (662),    in     proportion    as    this 
radius   is   greater,    and   as   the   radius   of 
the    eye    of    the   wheel    must    be    greater 

than  that  of  the  axle,  the  trunnion  has  the  advantage,  in   this   respect 
over   the  axle. 


APPLICATIONS.  420 

The  value  of  the  quantity  of  work  consumed  by  friction  is  wholly 
independent  of  the  length  of  the  trunnion  or  axle,  and  no  advantage 
is  therefore  gained  by  making  it  shorter  or  longer. 

J'3 

THE    CORD. 


§  362. — The  cord  and  its  properties  have  been  considered  in  part 
at  §  58.  It  is  now  proposed  to  discuss  its  action  under  the  opera- 
tion of  forces  applied  to  it  in  any  manner  whatever. 

Let  the  points  A',  A",  A'",  be  connected  with  each  other  by 
means  of  two  perfectly  flex- 
ible and  inextensible  cords 
A'  A'\  A"  A'",  the  first 
point  being  acted  upon  by 
the  forces  F',  P",  &c. ;  the 
second  by  the  forces  Q',  Q", 
&c. ;  and  the  third  by  the 
forces  S',  S",  &c. ;  and  sup- 
pose these  forces  to  be  in 
equilibrio.  Denote  the  co- 
ordinates of  A'  by  x'y'z', 
A"  by  x"  y"  z",  and  A'"  by 
x'"  y'"  z'".  Also,  the  alge- 
braic sum  of  the  components  of  the  forces  acting  at  A'  in  the  direc- 
tion of  X  y  2,  by  X'  Y'  Z\  at  A"  by  X"  Y"  Z" ,  and  at  A'"  by 
X"'Y"'Z"'.      Then  will,  §101, 

X'    5  x'    -f   Y'    Sy'    +  Z'    Sz'    ^ 
-\-  X"  6  x"    -f   Y"  S  y"    +  Z"  Sz"     [=  0.    '     '     (GG3) 
-f  X'"  S  x'"  +   Y'"  8  y'"  +  Z'"  S  z'"  J 

Denote  the  length  A'  A"  by  /,  and  A"  A'"  by  g ;    then  will 
L  :=.f-^'''-x'f+{y"  -y'Y  +  {z"  -z'y  =  0-  \ 

II  =r;-  VW"  -  ^'f  +  W"  -  y"f  +  (^'"  -  ^•")'  =  ^-     J 

The    displacement    by   which  we    obtain    the    virtual    velocities    whose 


4:80  ELEMENTS     OF     ANALYTICAL     MECHANICS. 

projections  are    8  x\  o  y\  S  z',  &c.,  is  not  wholly  arbitrary;    but  must 
be  made  so  as  to  satisfy  the  condition 

Sf=  0     and     5ff  =  0. (665) 

Differentiating    Equations    (664),    and    writing    for    dx',    d  y\    d  z\ 
6  x\  0  7j\  Ss',  &c.,  we  find 

{x"  -  x'){Sx"  -  5x')  +  jy"  -  ?/){Si/'  -  Si/)  +  {2"-s'){Sz"  -  §z')  _ 

f  -^' 

{x"'-x''){Sx'"-Sx")-\r{!/"-y"){Sy"'-Sy")  +  {z"'-z"){S2"'-Sz") 


These  being  multiplied  respectively  by  X'  and  X'",  and  added  to 
Equation  (663),  we  obtain  by  reduction,  and  by  the  principle  of 
indeterminate    co-efficients,  exactly  as  in  §213, 


X'  _  V.^^^-^'  =  0;  ^ 


/ 


y  -  X' . 


Z'  — X' 


,  y"-y' 


-."      „' 


0; 


0; 


(680) 


X"  +  X' 


r'"  —  r" 
X'".- ^=:0: 


Y"  +  X 


f  9 

II  „i  „iii  „;/ 


Z"  +  X' 


-  X' 


/  9 

jn  ^.11 


=  0; 


0; 


y  .      .      (667) 


X'"  +  X"'^- =0; 


v'"  —  v" 
T"  +  X'"  •  ^- ^  =  0 ;   }> 


^in   ^.11 

Z'"  +  X'" —  =  0 


(GGS) 


Taking  from   each   group  its  first   equation  and  adding,  and   doing 
the  same  for   the  second   and   third,  we   have 


X'  +  X"  +  X'"  =  0  ; 
Y'  +  Y"  +  Y'"  =  0 ; 
Z'  +  Z"  +  Z'"  =  0. 


{(iGd) 


APPLICATIONS. 


431 


That   is,    the    conditions    of    equilibrium  of    the  forces   are,    §  80,    the 
same    as   though   they  had   been    applied  to    a   single   point. 

To  find  the  position  of  the  points,  eliminate  the  factors  X'  and 
A.'",  and  for  this  purpose  add  the  first,  second  and  third  equations 
of  group  (667)  to  the  corresponding  equations  of  group  (668),  and 
there  will    result 

X"  +  X'"  +  J  {x"  -  x')  =  0  ; 

Z"+    Y'"  -\-  J  {,/'   -  y')    =   0- 

Z"  +  Z"'  +  A  (2"  _  2')  ^  0. 
from  which  we  find  by  elimination. 


Y"  +  Y'"  -  ^, ^,  (X"  +  X'") 

X       —   X 


'"-  {X"  +  X'")  =  0. 


(070) 


Z"  +  Z'"  -  -^ 
Fi'om    group  (666)5  ^Y  eliminating  X', 

y    —  y 


Y' 
Z' 


X'  =  0 


X'  =  0 


(671) 


and    finally  fium    group   (668)  we   obtain,  by    eliminating   X'", 

v'"  —  v" 

^         ~  1777 177  •  -^        —  "  : 


Z'"  - 


X'"  =  0. 


(672) 


Equations  (669),  (670),  (671)  and  672),  involve  all  the  conditions 
necessary  to  the  equilibrium,  and  the  last  three  groups,  in  connection 
with  group  (664),  determine  the  positions  of  the  points  A',  A" 
and  A'",  in  space. 


303. — The    reactions    in    the    system    which   impose  conditions  on 


432  ELEMENTS     OF     ANALYTICAL    MECHANICS. 

the    displacement   will     be    made    known    by    Equation    (331),   whiclj, 
because 

[jC^^^T^J  +L^n7^"7)J  "^L^F^"^]  "^' 

[d{x"'-x")j   ^ld{!j"'-y")j   ^  \jl  [z'"  -  ,")j   -  '' 
becomes  for  the   cord  A'  A", 

■k'  =  JV'; 
and  for  the  cord  A"  A'", 

-k'"  =  N'"  ; 

from  which  we   conclude,  that   X'  and   X'"   are  respectively  the   ten- 
sions of  the   cords  A' A"  and  A"  A'". 

This    is    also    manifest  from    Equations  (GGG)    and    (668)  ;    for,   by 
transposing,  squaring,  adding  and   reducing    by  the    relations, 

(.,-  _  .^.')2  +  (y-  -  ,jr  4-  jz"  -  z'Y   _ 

—  *■■> 


P 

(^x'"  -  x"f  +  {v'"  -  y"Y  +  {z'"  -  z"Y 

9" 
we   have 


=  1, 


X'     =    -y/X'-^       +    ¥'•'       +    Z'2       =    R'^ 


(673) 


in  Avhich  i^'  and  R'"  are  the  resultants  of  the  forces  acting  upon 
the  points  A'  and   A'"  respectively. 

Substituting   these   values   in    Equations  (GGG)  and  (6G8),  we  have 

X'  _  x"  -  x'  _      Y^  _  y"  -  y'  ^      Z/^  _  z"  -  z'  ^ 
Rf  ~        f        '      R'  ~        f       '     R'  ~        7       *' 

X'"  x'"  —  x"      T"  _        y'"  —  y"      Z'"  z'"  —  z" 

whence  the  resultants  of  the  forces  applied  at  the  points  A'  and  y1'", 
act  in  the  directions  of  the  cords  connecting  these  points  with  the 
point  A",  and  will  be  equal  to,  indeed  determine  the  tensions  of 
these   cords:; 


APPLICATIONS.  433 

§  364. — From    Equations    (669),    we    have   bj    transposition, 

X"  =  -  {X'"  +  X')  ■    Y"  =  -  {¥'"  +  Y') ;    Z"  ^  -  [Z'"  +  Z'). 

Squaring,    adding   and    denoting   the    resultant   of    the   forces   applied 
at  A"  by  R",  we  have 


E"  =z  y/{X"'  +  X'f  +  (F'"  +  Y'f  +  (Z'"  +  Z'f  ■  .  (674) 
and    dividing  each  of  the   above    equations   by  this  one 

X'"  +  X'     ^ 


X" 
R" 

Y" 
R" 

Z" 

~Rr 


R" 

Y'"  + 

Y' 

R" 

Z'"  + 

Z' 

R" 


(675) 


-^"x 


whence,  Equation  (674),  the  resultant  of  the  forces  applied  at  A"  is 
equal  and  immediately  opposed  to  the  resultant  of  all  the  forces 
applied   both  at  A'  and  A'" 

If,  therefore,  from  the  point 
A"^  distances  A"  vi  and  A"  n 
be  taken  i^roportional  to  R'  and 
R'"  respectively,  and  a  paral- 
lelogram A"  m  On  be  constructed, 
A"  G  will  represent  the  value  of 
R".  If  A'  A"  A'"  be  a  contin- 
uous cord,  and  the  point  A" 
capable  of  sliding  thereon,  the 
tension  of  the  cord  would  be 
the  same  throughout,  in  which 
case  R'  would  be  equal  to  i2"', 
and  the  direction  of  R"  would 
bisect  the  angle  A'  A"  A'". 

The  same  result  is  shown  if, 
instead    of    making    (J/  =  0    and 

5  y  =  0    separately,    we     make 

28 


A  " 


4:34         ELEMENTS     OF    ANALYTICAL    MECHANICS. 

S  (/  |-  ff)  =  0,     multiply    by    a     single     indeterminate     quantity    X, 
and  proceed    as    before. 

§  365. — Had  there  been  four 
points,  A',  A",  A"'  and  A^^ 
connected  by  the  same  means, 
the  genei'al  equation  of  equili- 
brium would  become,  by  call- 
ing h  the  distance  between  the 
points,  A'"  and  A^"^, 

X'  S  x'  +  X"  S  x"  +  X"  5  x'"  -I-  X^''  5  a;i^  "1 

-f  T'  8tj'  +  Y"  S  tj"  -f  Y'"  6  y'"  +  Y^^  8  ?/i 

+  Z'  5  2'  +  Z"  S  z"  +  Z'"  S  z'"  +  Zi-  8  z^ 

+  V  5/  +  X"  <J  ^     +  y^'"  S  h 


}  -0; 


and    from  which,  by  substituting    the  values  of  S  f^  8  g    and  8  h,  the 
folic  wing  equations  will  result,  viz.  : 


X  -X'  • 


f 


Y'-X'.  ^L-=L1L  ^  0. 


X'  +  X' 
Y"  +  X- 


7'          'X'  . 

—  0 

f 

—  u, 

x"  -  x' 

-  -k" 

x'"  -  x" 

f 

,   y"  --  y' 


f 


Z"  +  X' 


X"  -f  X" 


Y'"  +  X 


Z'"  +  X" 


//  y 


f 

x'"  - 

x" 

9 

y'"  - 

y" 

9 

z'"   - 

^11 

-  X' 


-  X' 


9 

y'"  -  y' 

9 

^111  g' 


0, 
0, 
0, 


—  X 


—  X' 


0, 
0, 
0, 


(C7G) 


(677) 


V     .     (078) 


APPLICATIUNS. 


435 


,.iv   .,.'" 


Xi^  +  >^"' 

0 

h     -^' 

jnv  ^  X'" 

2iv     +   V" 

,iv   __    ,r. 

h     -  ^'  I 

(679) 


Eliminating  the  indeterminate  quantities  X'  X"  and  X'",  we  obtain 
eight  equations,  from  which,  and  the  three  equations  of  conditions 
expressive  of  the  lengths  of/,  g,  and  A,  the  position  of  the  points  A', 
A",  A"\  and  A^"  may  be  determined. 

If  there  be  n  points,  connected  in  the  same  way  and  acted  upon 
by  any  forces,  the  law  which  is  manifest  in  the  formation  of  Equa- 
tions (676),  (677),  (678),  and  (679),  plainly  indicates  the  following 
n  equations  of  equilibrium : 


X'  -\'  .  ^    ~^'   =  0, 


F  -  X'  . 


,  y"  -  y 


f 


=  0, 


Z'    -  X'  .  -"-—;—-  =  0, 


(680) 


j't        n.11 


X"  +  X'  .  ^     ,  ""   -  X"  . =  0, 


/ 


r"  +  X 


>  y"  -  y'    ■."  y'"  -  y" 


f 


=  0, 


ji     -/ 


,.111 ^" 


Z"  +  X' . '— =-  -  X"  .  ^_zi^  =  0, 
/  9 


(A-n) 


^rn  ^,11 


X"  +  X"  . ""'"  ~  •^"  -  X'"  .  ^ ^  =  0, 


Y'"  4-  X 


„   y"'-y" 


-  X' 


2"'  +  X"  . X' 


h 

yiv_ 

y'" 

h 

giy   _ 

=  0, 
=  0, 


(682) 


4-36  ELEMENTS     OF    AISALYTICAL    MECHANICS. 


X„_i  +  X„_2 


^n-l     +    K-, 


«.-! 

—    ^n-2 

k 

y.-i 

—    Vn-^ 

k 

Zn-\ 

—     2„-2 

^r,  —  a'„ 


V           _!_■>,                 :/n-i             ;/n--i             ^                 j/n           l/n-l  /^ 

-r„_l   +    A„_2  • A„_i   • =:    U, 


K-,  ■  '-^^-^^  =  0, 


(683) 


x„  +  x,,,.^"   ;''"-' :^0, 


i^„  +  x„..-^^^  =  o, 


^n     +5.„^i 


Z 


=  0. 


(684) 


In  which  X,  with  its  jDarticular  accent,  denotes  the  tension  of  the 
cord  into  the  difference  of  whose  extreme  co-ordinates  it  is  multi- 
plied. 

Adding  together    the   equations   containing   the   components  of  the 
forces  parallel  to  the  same  axis,  there  will  result 


X'  +  X"  +  X'"  +  A'i^ 
Y'  +  Y"  +  Y'"  +  Y^' 
Z'   +  Z"  +  Z'"  +  Z'^ 


r„  =  0,    I  •     •     (685) 
^»  =  0,  J 


from    Avhich   we    infer,    that     the    conditions    of    equilibrium    are   the 
same  as  though  the  forces   Avere  all   applied  to  a  single  point. 

From    group  (680),  we   find   by  transposing,  squaring,  adding  and 
extracting    square   root, 


yA"2  +  Y'-'  +  Z'^  =  X'  =  E' 

and  dividing  each  of  the  equations  found  after  transposing   in   group 
(680)  by  this   on  3, 

X'        x"  -  x' 


R' 

f 

T 

= 

/' 

— 

y' 

B' 

f 

Z' 

= 

z" 

— 

z' 

E' 

f 

APPLICATIONS. 


437 


Treating    the    equations  of    group    (684)    in    the     same    way,    wo 
have 


Jin 


x„ 

—  «»-! 

I 

Vn 

-Vn-X 

I 

^n 

—  Zn-X 

I 


whence,  the  resultants 

of  the   forces    applied 

to  the  extreme  points 

A'  and  A,^,  act  in  the 

direction  of  the  extreme  cords.     And  from  Equations  (G85)  it  appears 

that   the   resultant   of  these   two  resultants   is  equal   and   contrary  to 

that    of  all   the    forces    applied   to    the    other   points. 

§366.— If  the  extreme  points  be  fixed,  X\  V,  Z'  and  X^,  F„,  Z„, 
will  be  the  components  of  the  resistances  of  these  points  in  the 
directions  of  the  axes ;  these  resistances  will  be  equal  to  the  ten- 
sions \'  and  X„  of  the  cords  which  terminate  in  them.  Taking  the 
sum  of  the  equations  in  groups  (680)  to  (684),  stopping  at  the  point 
whose    co-ordinates   are  x,^_,^^  y,i-mi  ^n-mt  we   have 


Z'  -{-  ^  Z  —  X,^„  —^ 


^.-m 

—    Xn-m-l 

1 

L-,. 

Vn-u 

—    yn-m-l 

' 

4-« 

^n-rr 

—    2«-m-l 

=  0; 

=  0 


J 


(686) 


in  which  2  X,  2  Y,  2  Z,  denote  the  algebraic  sums  of  the  components 
in  the  directions  of  the  axes  of  the  active  forces;  X„_„^_^  the  tension 
on  the  side  of  which  the  extreme  co-ordinates  are  a;„..„,  y„..,„,  2^„, 
and  :r„_„_i,  y„-,„.-i,  2«-m-i;   and  Z„_„  the  length  of  this   side. 


367, — Now,    suppose   the    length   of    the    sides    diminished    and 


438         ELEMENTS     OF     ANALYTICAL     MECHANICS. 


their    number    increased     indefinitely ;    the    polygon    will     become    a 
curve;   also,  making  X,^„_j  =  ;,  we   have 

y«-m  —  y»-»^i  —  dy, 

/„_„  =z  d  s, 
s  being   any   length  of  the    curve  ;  and  Equations  (686)  become 

X'  +  2  A' 

dy 


dx 

t.-y-     =    0 

a  s 


r  +  2  F  -  ^ 


ds 
dz 


0;  \ 


Z'  +  2Z-<-—  =0; 
d  s 


(687) 


which  will  give  the  curved  locus  of  a  rope  or  chain,  ftistened  at 
its  ends,  and  acted  upon  by  any  forces  whatever,  as  its  own  weight, 
the  weight  of  other  materials,  the  pressure  of  winds,  currents  of 
water,  &c.,  &c. 

This  arrangement  of  several  points,  connected  by  means  of  flexi- 
ble cords,  and  subjected  to  the  action  of  forces,  is  called  a  Funi- 
cular Machine. 

§3(58. — If  the  only  forces  acting  be  pressure  from  weights,  we 
have,  by  taking    the   axis   of  z  vertical, 

X"  =  X"  =  X'  &c.  =z  0 ;     F"  =  Y'"  &c.  =  0 ; 

and    from   Equations  (680)   to   (684), 


X'  =  X' 


/ 


Xfl-l  — 


whence,  the  tensions  on  all  the  cords,  estimated  in  a  horizontal 
direction,  are  equal  to  one  another.  Moreover,  we  obtain  from  the 
same    equations,    by  division. 


f"  -  y 

x'"  -  x' 


X.i    —  X,^\ 


APPLICATIONS. 


439 


These  are  the  tangents  of  the  angles  which  the  projections  of  the 
sides  on  the  plane  xij  make  with  the  axis  x.  The  polygon  is 
therefore   contained  in   a  vertical  plane. 

THE   CATENAEY. 


s-y 


§369.— If  a  single  rope  or  chain  cable  be  taken,  and  subjected 
only  to  the  action  of  its  own  weight,  it  will  assume  a  curvilinear 
shape  called  the  Catenary  curve.  It  will  lie  in  a  vertical  plane. 
Take  the  axes  z  and  x  in  this  plane,  and  z  positive  upwards,  then 
will 

2X=0;     2F=:0;      Y'  =  0 ;     2Z=-W; 

in    which   W  denotes   the  weight  of  the   cable,  and    Equations  (687) 
become 


CIS 


0, 


dz 
W  -i-—  =0. 

d  s 


y.'s  --  -A  y^ 


(688) 


These    are    the    differential    equations   of    the    curve.     The    ori"-in 
may  be   taken  at  any  point.      '^^   ^.y  t.-cuc-i^^.,^    ,,,     vi-uU^iv   *^i^^.''. 
Let  it  be  at  the  bottom  point     <^*^^  "^  "^riT-^  ^   *^  '^  UytJ} f<^^ 
of    the     curve.       The    curve 
being   at    rest,    will    not    be 
disturbed  by  taking  any  one 
of  its   points  fixed    at  pleas- 
ure.      Suppose     the     lowest 
point   for   a   moment   to    be- 
come   fixed.      As    the    curve 

is   here  horizontal,  Z'  =  0,  §  366,  and  from   the   second  of  Equations 
(688),  we   have 


ds 


(689) 


whence,  the  vertical  component  of  the  tension  at  any  point  as  0  of 
the  curve,  is  equal  to  the  weight  of  that  part  of  the  cable  between 
<his  point  and  the  lowest  7X)int.     The  first  of  Equations  (688)  shows 


410  ELEMENTS     OF    ANALYTICAL    MECHANICS. 

that  the  horizontal  component  of  the  tension  at  0  is  equal  to  the 
tension  at  the  lowest  point,  as  it  should  be,  since  the  hor-zontal 
tensions   arc   equal  throughout. 

Taking  the  unit  of  length  of  the  cable  to  give  a  unit  of  weight, 
which  would  give  the  common  catenary,  we  have  W  =z  s )  and,  de- 
noting  the   tension  at   the  lowest   point  by  c,  we  have 


and  from  Equation  (089), 

_      s  •  ds 


Taking  the  positive  sign,  because  z  and  s  increase  together,  inte- 
grating, and  finding  the  constant  of  integration  such  that  Avhen 
2  =z  0,  we   have   s  =  0, 


whence,  "Z-  -   V  c'-tS   — 

s2   =   22   _|_   2c2. 

Also,  dividing  the  first   of  Equations  (G8S)  by  Equation  (089), 

dx  c  c 

d-  ^    s    ~    y^22  +  2cs' 

and   integrating,  and   taking   the   constant   such   that   x   and   z  vanish 
together, 


X  =  c-log    — ^- — ^^—^ •      •      .     (090) 

which   is    the    equation  of  the  catenary. 

This   equation   may   be   put    under    another   form.      For  we   may 
write   the   above, 


transposing  z  -{-  c  and    squaring, 

c^ .  e~  —  2ce^  {z  +  c)  =  —  c^  ; 
whence, 

2  +  c  =  -ic-(e^  +  e~^). (691) 


Kip-d^'-^uYf-  '/.^)-c»oi/.^z^^/-^ 


APPLICATIONS. 


441 


Also, 


=  V(.^  +  cf  - 


and   by    substitution, 

s  =  ^  c  •  (e~  —  e~"). (092) 

§3Y0. — If  the  length  of  the  portion  of  the  cable  which  gives  a 
unit  of  -weight  were  to  vary,  the  vaiiation  might  be  made  such  as 
to  cause  the  area  of  the  cross  section  to  Ije  proportional  to  the 
tension  at  the  point  where  the  section  is  made.  The  general  E(pia- 
tions  (688)  will    give    the    solution   for    every  possible  case. 


FRICTION   BETWEEN    COEDS    AND    CYLINDEICAL    SOLIDS. 

§  371. — When  a  cord  is  wrapped  around  a  solid  cylinder,  and 
motion  is  communicated  by  applying  the  power  F  at  one  end 
while  a  resistance  W  acts  at  tlie  other,  a  pressure  is  exerted  by 
the  coi'd  upon  the  cylinder  ;  this  pressure  produces  friction,  and  this 
acts  as  a  resistance.  To  estimate  its  amount,  denote  the  radius 
of  the  cylinder  by  i?,  the  arc  of  contact  by  s,  the  tension  of  the 
cord    at   any  point  by  t. 

The  tension  t  being  the  same 
throughout  the  length  d  s  z=z  a  t^ 
of  the  cord,  this  element  will  be 
pressed  against  the  cylinder  by 
two  forces  each  equal  to  t,  and 
ajiplied  at  its  extremities  a  and  i^ , 
the  first  acting  from  a  towards 
W,  the  second  from  7^  towards  b'. 
Denoting  by  6  the  angle  a  b  t^ , 
and  by  p  the  resultant  b  m  of 
these  forces,  which  is  obviously 
the    pressure  of  els  against   the   cylinder,  we   have.  Equation  (50), 


but 


2)  =  y/t-  +  t^  +  2  ;; . ;  cos  0  =  t  ■\/'l{\  +  cos  ^)  ; 


JLJ^*} 


1  +  cos  ^  =z  2  cos2  I-  ^  •,  (tt  —  ^)  =  — -  ; 


uv^-f- 


..UO^^L^ 


^rrf-) 


442  ELEMENTS     OF    ANALYTICAL    MECHANICS. 

and   taking   the   arc   for    its    sine,    because   tt  —  d   is   very   small,    we 
have 

d  s 

and   hence,  §355,  the  friction  onds  will  be 

The  element  t^  t^  of  the  cord  which  next  succeeds  at^^  will  have 
its  tension  increased  by  this  friction  before  the  latter  can  be  over- 
come ;  this  friction  is  therefore  the  differential  of  the  tendon,  being 
the    difference    of  the   tensions  of  two  consecutive   elements  ;  whence, 

ds 
dt=f't.-; 

dividing   by   t   and   integrating. 


or. 


log  ^  =  /•  ^  +  ^^g  ^' 


t  =  Ce'' (693) 

making  s  =  0,  we   have  t  =i  W  =   C ;  whence, 

t  =z  W-  eJ^  ; (694) 

and   making  s  =  ^S  =  a^i  ^2  4,  we   have  t  =  F;  and 

F=  W-e^       (695) 

Suppose,  for  example,  the   cord   to   be  wound  around  the  cylinder 
three  times,  and  f  =  ^  ;  then  will 

^^  r=  3  *  .  2  /^  =:  G  .  3,]41G  .  i2  =  18,849  E, 
and 

i^=Trx.^^^^'^"^=Trx  (2,71825)^'^; 
or, 

F=  IF.  535,3- 

that  is  to  say,  one  man  at  the  end  TV  could  resist  the  combined 
efibrt  of  535  men,  of  the  same  strcnjrih  as  himself  to  put  the  cord 
in    molioii   w};e!i    wound    three    tiiries    loiuid    the    (-vliiule]. 


APPLECATIONS. 


443 


THE    IXCLIXED    PLAXE. 

§  372. — The  inclined  plane  is  used  to  support,  in  part,  the  weight 
of  a    body  while    at    rest  or    in    motion    upon    its    surface. 

Suppose  a  body  to  rest  with  one  of  its  faces  on  an  inchued  plane 
of   which    the    Equation    is 

L  =  cos  a X  -\-  coib  y  -\-  CQi c z  -—  cl  =^  0  ;  •  ■  •  •  (a) 
in  which  d  denotes  the  distance  of  the  plane  from  the  origin  of  co- 
ordinates, and  a,  b,  c,  the  angles  which  a  normal  to  the  plane  makes 
with    the    axes   .r,  y,  z,  respectively. 

Denote  the  weight  of  the  body  by  W ,  the  power  by  F ;  the  nor- 
mal pressure  by  N ;  the  angles  which  the  power  makes  with  the 
axes  X,  y,  z,  by  a^,  /3^,  y^,  respectively;  and  the  path  described  by 
the  point  of  application  of  the  resultant  friction  by  s.  Then,  taking 
the  axis  z  vertical  and  positive  upv.ard.s,  and  supposing  the  force  to 
produce    a  uniform    motion  of  simple   translation,  will,  Eq.  (G45), 

(i^cos.^+/iXr^)^. 


+  (i^cos/3^+/iV^-[^^)<)y 

+  (fcos  7;  +/iV  ^^  -  if)  0  z 


0; 


(^) 


and,    Equation    (a), 

cos  aox  -\-  cos  liS  y  -\-  cos  c  o  z  =  0 
Multiplyhig     this    last    by    a,    adding    and    proceeding    as   in    §  213. 


F  cos  a^  -{-  f  N  — -  +  ^  cos  a  —  0, 

F  cos  /3 .  +  /  i\^  ^  +  X  cos  6  =  0, 

as 

Fcosy,  +  /  JV^  -^  +  X  cos  c  -  TF  =  0  ; 


iO 


and,  Eq.  (331), 


■  + 


c^)  +  (- 


;    )  =x. 

dx/  ^dy/  \dz/ 


id) 


Substituting  the  val  le  of  X  in    Equations  {c),  tiie    first    two  give  by 
eliminating    N, 


4:4:4: 


ELEMENTS  OF  ANALYTICAL  MECHANICS. 


f f-  cos  a 

as 

^  dv 

f  ~  +  co%b 
as 


cos  /3^ 


+  1  =  0; 


ie) 


and    the   first   aud   third,  bv  eHniinating  N, 


lAfl cosy  — cosa^— ^ j-f-cos7^cos«  — cosa  cose    =  Wif—-\-co?:a\. 


is) 


If  there    be    no    friction,  then    will  y=  0,  and,  Eq.    (e), 


cos  a    cos  p^ 


+  1-  =  0  ; 


cos  6    cos  «^ 

whence,  Eqs.    (45)    and    («),    the    power    must    be    applied    in    a    plane 
normal    both   to    the    inclined   plane    and    to  the    horizon. 

If  without  disregarding  friction,  the  power  be  applied  in  a  plane 
fulfilling  the  above  condition,  and  also  con- 
taming  the  centre  of  gravity,  the  resultant 
friction  may  be  regarded  as  acting  in  tliis 
plane,  and  we  may  take  it  as  the  co- 
ordinate   plane    z  x,    in    which    case 

^         ^     d  y 
cos  6  =  0  ;  cos  /S,  =  0  ;  --  =  0  ; 
d  s 

and   denoting   the    inclination    of  the    plane    to   the   horizon  by   a,   and 
that    of  the    power    to    the   inclined    plane    by  (p  ; 

cos  a  =  sin  a  ;  cos  c  =:  —  cos  a  ;  cos  y^  =  sin  a.^ ; 

dx  dz  . 

cos  y I  — cos  a. I  —  =  —  sm  a^  cos  a  +  cosk^  sin  a  =  sin  (a—  ajr=sm  d  ; 

Ct  o  CL  Ki 

cos  y^  cos  a  —  cosa^  cosc  =  sm  a^  sin  a  +  cosa^  cosc.  =  cos(a  — cj^rccsyi; 
which,  in   Eq.  [g),  give 

„        W  (sin  a  +  /cos  a)  .        , 

F—  5^ '-^ J. (69(J) 

cos  (p  -f-  /  sin  9 

This  supposes  motion  to  take  place  itp  the  plane ;  if  the  power  F 
be  just  sufficient  to  permit  the  body  to  move  uniformly  down  the 
plane,  then  will  /  change  its  sign,  and  we  shall  have 

^^JF(sma_-/cos«)_ 

cos  (p  —  /  sui  9  ^       ' 

And  the  power  may  vary  between  the  limits  given  by  these  two 
values  without  moving  the  body. 


APPLICATIONS.  415 

§  373,  If  the  i^cwer  be  zero,  or  i^  =  0,  then  will 

sui  a  —  /  cos  a  =  0, 
or 

tan  a.  z=z  f^ 

which  is  the  angle  of  friction,  §  355. 

§  374. — If  the  power  act  parallel  to  the  plane,  then  will  9  =  0, 
and 

F—  TF(sin  a  ±/cos  a) (698) 

the  upper  sign  answering  to  the   case  of  motion   up,  and   the   lower, 
down  the  plane;    the  difference  of  the  two  values  being 

2/  IF  cos  a. 
If  /  =  0,   then   will 

F         .  EC 

that  is,  the  power  is  to   the  weight  as  the  height  of  the  plane  is  to 
its  length;    and   there   will   be  a  gain  of  power. 

§375. — If  the  power  be  applied  horizontally,  then  will  9  be  nega- 
tive and  equal  to  a,  and  we  have,  by  including  the  motion  in  both 
directions, 

^  ^   Wisma^fcos^) 

cos  a  q:  /  sin  a  ^       ' 

the  difference  of  the  limiting  values  being 

2/.  W 


COS^  a   —  /2  sjn2  a 

If  the  friction  be  zero,  or  /  =  0,  then  will 

F  BO 

-  =  tana  =  ^. 

That    is,  the    power  will    be    to    the    resistance    as    the  height  of  the 
pla-no  is  to  its  base;    and    there   may    be  gain    or   loss   of  power. 

§  376. — To  find  under  w^at  angfe  the   power   will  act   to   greatest 
advantage,   make    the    denomhiator    in    Equation    (696)    a   maximum. 
For  this  purpose,  we  have,  by  differentiating, 
—  sin  9  +  /cos  9  ::=  0  ; 


44G 


ELEMENTS     OF    ANALYTICAL    MECHANICS 


whence, 

tan  <p  =  /. 

That  is,  the  angle  should  be   positive,  and  equal  to  that  of  the  fric- 
tion. 

g  377. — If  the  power  act  parallel  to  any  inclined  surface  to  move 
a  body  up,  the  elementary  quantity  of  work  of  the  power  and  resist- 
ances will  give  the  relation.  Equation  (698), 

F d  s  =  Wds  sin  a  +  Wfd  s  cos  «. 

But,  denoting  the  whole  hori- 
zontal distance  passed  over  by 
I  =  A  C,  and  the  vertical  height 
by  h  =  £  C,  we  have 

d  s  .  sin  a  =  d  h, 


d  s  .  cos  a  =  d  I: 


A    0 


whence,  substituting,  and  integrating,  and  supposing  the  body  to  be 
started  from  reet  and  brought  to  rest  again,  in  which  case  the  work 
of  inertia    will    balance  itself,  we  have 

Fs=Wh+f.W.l, (700) 

in  which  there  is  no  trace  of  the  path  actually  passed  over  by  the 
body.  The  work  is  that  required  to  raise  the  body  through  a  ver- 
tical height  B  C.  and  to  overcome  the  friction  due  to  its  weight  over 
a   horizontal    distance   A  C. 

The  resultant  of  the  weight  and  the  power  must  intersect  tne 
inclined  plane  within  the  polygon,  formed  by  joining  the  points  of 
contact  of  the  body,  else  the  body  will  roll,  and  not  slide. 


b'^^ 


THE    LEVEK. 


§  378. — The  Lever  is  a  solid 
bar  A  B,  of  any  form,  supported 
by  a  fixed  point  0,  about  which 
it  may  freely  turn,  called  the  ful- 
crum. Sometimes  ifc  is  supported 
upon     trunnions,    and    frequently 


APPLICATIONS. 


447 


upon  a  knife-edge.  Levers  have 
been  divided  into  three  different 
classes,  called  orders. 

In  levers  of  the  first  order,  the 
power  F  and  resistance  Q  are 
applied  on  opposite  sides  of  the 
fulcrum  0\  in  levers  of  the  second 
order,  the  resistance  Q  is  applied 
to  some  point  between  the  ful- 
crum 0  and  the  point  of  appli- 
cation of  the  power  F  \  and  in 
the  third  order  of  levers,  the 
power  F  is  applied  between  the 
fulcrum  0  and  point  of  applica- 
tion of  the  resistance  Q. 

The  common  shears  furnishes 
an  example  of  a  pair  of  levers 
of  the  first  order  ;  the  nut-crackers 
of  the  second ;  and  fire-tongs  of 
the  third.  In  all  orders,  the  con- 
ditions of  equilibrium  are  the 
same. 

These  divisions  are  wholly  ar- 
bitrary, being  founded  in  no  dif- 
ference of  principle.  The  relation 
of  the  power  to  the  resistances, 
is   the  same   in  all. 

Let  ^1  J5  be  a  lever  supported 
upon  a  trunnion  at  0,  and  acted 
upon  by  the  power  P  and  resist- 
ance Q,  applied  in  a  plane  per- 
pendicular to  the  axis  of  the  ti'un- 
nion.  Draw  from  the  axis  of  the 
trunnion,  the  lever  arms  0  n  and 
0  m,  being  the  perpendicular  dis- 
tances of  the  power  and  resistance 
fi'ou"     tlie    axis    of     m(jti>jii,    and 


F 


1% 


44S  ELEMENTS     OF    ANALYTICAL    MECHANICS. 

denote    them   respectively  by  Ip  and  I, ;  also    denote  the   resultant  of 
P  and   Q  by  N,  the   radius  of  the   trunnion   by  r,  the  co-efficient  of 
friction   by  /,  and   the   arc   described   at  the  unit's   distance  from   the 
axis   by  Sj . 
Then, 

8  p  =  Ip  .  d  s^  ;      S  q  z=  l^ .  d  Si  . 


N  =^  ^P"  4-  i^'  -^  'ZF  (^  cos  t), 

in  which  ^  is  the  angle  of  inclination  A  C  B  of  the  power  to  the 
resistance.  Then,  supposing  the  lever  to  have  attained  a  uniform 
motion,  will,  Equations  (645)  and  (661), 

r  .  d  Si ./ 


P.L.ds,-Q.K.ds,--y/P-'+  §2  4_op^^,,,d.—^=^i^z=  0.(701) 
Omitting  the   common  foctor  f/s,,  and  making 


/  _.v.       .„_     I. 


=  y  ;    "i  ^  r  5    '*  = 


r 


we  have, 


P  ^m  Q  -  -/P3T~^M^'2P~^T^^^  -f'n  =  0. 
Transposing,  squaring,  and  solving,  with  respect  to  P,  we  find, 


•;//  +  r'  n  ( /'  n  os  ()  ±  -/!  +'Zm  cos  a  +  in-  —  /'2  n2  sin2  d) 

If  the  fraction  n  be  so  small  as  to  justify  the  omission  of  every 
term  into  which  it  enters  as  a  factor,  or  if  the  co-efficient  of  friction 
be  sensibly  zero,  then  would 

|="'=i ('°^> 

That  is,  the  power  and  the  resistance  are  to  each  other  inversely  as 
the  lengths  of  their  respective  lever  arms. 

If  the  power  or  the  resistance,  or  both,  be  applied  in  a  plane 
oblique  to  the  axis  of  the  trunnion,  each  oblique  action  must  be 
replaced  by  its  components,  one  of  which  is  perpendicular,  and  the 
other  parallel  to  the  axis  of  the  trunnion.  The  perpendicular  com- 
ponents must  be  treated  as  above.     The  parallel  components  will,  if 


APTLICATIONS. 


449 


the  friction  arising  from  the  resultant  of  the  normal  components  be 
not  too  great,  give  motion  to  the  whole  body  of  the  lever  along  the 
trunnion  ;  and  if  this  be  prevented  by  a  shoulder,  the  friction  upon 
this  shoulder  becomes  an  additional  resistance,  whose  elemcntarv 
quantity  of  work  may  be  computed  by  means  of  Eq.  (CSV)  and  made 
another  term  in  Equation  C^Ol). 


WHEEL   AND   AXLE. 

§  3 7 9. — This  machine  consists  of  a  wheel  mounted  upon  an  arbor, 
supported  at  either  end  by  a  trun- 
nion resting  in  a  box  or  trunnion 
bed.  The  plane  of  the  wheel  is  at 
right  angles  to  the  arbor ;  the  pow- 
er P  is  applied  to  a  rope  wound 
round  the  wheel,  the  resistance  to 
another  rope  wound  in  the  opposite 
direction  about  the  arbor,  and  both 
act  in  planes  at  right  angles  to  the 
axis  of  motion.  Let  us  suppose  the 
arbor  to  be  horizontal  and  the  re- 
sistance  §  to  be  a  weight. 

Make 
iV  and  N'  =  pressures  upon  the  trunnion  boxes  at  A  and  B ; 
R  =z  radius  of  the  wheel ; 
r  =  radius  of  the  arbor ; 
p  and  p'  =  radii  of  the  trunnions  at  A  and  B ; 


s,  ==  arc    described    at    unit's  distance    from  axis  of   motioa 
Then,  the  system  being  retained  by  a  fixed  axis,  we  have 

P  ^p  =  PRds,; 
Q  S  q  z=  Q  r  d  Si. 
The  elementary  work  of  the  friction  will,  Eq.  (G6l),  be 

29 


450  ELEMENTS     OF    ANALYTICAL    MECHANICS. 

and    the    elementary    work    of    the     stiffness   of    cordage,    Equation 
(652), 

K-\- 1  .q 

d,  . — ■  r  '  ds^\ 

and  when  the  machine  is  moving  uniformly, 

PRds^-qrds^-f{N^^I^'^')ds,-d^--^^^.r'ds,  =  ();-  ('704) 

The  pressures  N  and  N'  arise  from  the  action  of  the  power  P,  the 
weight  of  the  machine,  and  the  reaction  of  the  resistance  Q.  in- 
creased by  the  stiffness  of  cordage.  To  find  their  values,  resolve 
each  of  these  forces  into  two  parallel  components  acting  in  2:)lanes 
which  are  perpendicular  to  the  axis  of  the  arbor  at  the  trunnion 
beds ;  then  resolve  each  of  these  components  which  are  oblique  to 
the  components  of  Q  into  two  others,  one  parallel  and  the  other 
perpendicular  to  the  direction  of  Q. 
Make 

to    =  weight    of  the    wheel   and   axle, 

g    =  the   distance   of  its   centre   of  gravity  from  A, 

p    =  the    distance   m  A, 

q    =  the    distance   n  A, 

I    =  length   of  the   arbor   A  B, 

(p    =z  the   angle   which   the   direction   of  F  makes   with   the    vertical 

or    direction    of    the    resistance    Q. 
Then    the    force    applied    in    the   jilane  perpendicular    to    tlie    trunnion 
A,  and  acting  parallel  to  the  resistance   Q,  will,  §  95,  be, 

io ~  +   Q ^ h  P J cos  (p  ; 

and  the  force  applied  in   this  plane   and  acting  at  right  angles  to  the 
direction  of  Q,  will  be 

F —  •  sm  9, 

The  vertical  force  applied  in  the  plane  at  B  will  be 


ATPLICATIOlSrS.  451 

and  the  horizontal  force  in  this  plane  will  be 

P  .  I  .  sin  9  ; 
whence, 

iV  =  1  V  \w{l-g)+Q(l-q)  +  P{l-pYo^^^■i-{-P\l-pY.Bm^(^  ■  .  (705) 

JSf'=j.-^[iu.ff  +   Q  .q  +  P.i?.cos(p]2  _^  p-^.j,^  .sm^(p',  ■   •  (70G) 

If  6  and  6'  he  the  angles  which  the  directions  of  iV  and  JV'  make 
with  that  of  the  resistance   Q,  we  have 

.     ^        P{l-p)      .  .     ^,        Pp      . 

sin  ^  =  — ^^^ sui  (p  ;     sni  ^'  -  ^^  •  sm  9. 

Equations  (704),  (705),  and  (70G)  are  sufficient  to  determine  the  rela- 
tion between  P  and  Q  to  j)reserve  the  motion  uniform,  or  an  equili- 
brium without  the  aid  of  inertia.  The  values  of  iV"  and  N'  being 
substituted  in  Equation  (704),  and  that  equation  solved  with  refer- 
ence to  P,  will  give  the  relation  in  question. 

§  380. — If  the  power  P  act  in  the  direction  of  the  resistance  Q, 
then  will  cos  9  =:  1,  sin  9  =  0,  and  Equation  (704)  would,  after 
substituting  the  corresponding  values  of  N  and  N\  transposing, 
omitting   the    common    flictor  d  s^ ,   and    supposing   p  z=  p',   become 

PB=  Qr  -\-f'^{w  +  (2  +  P)  +  d^ .  -   \      ^  -r.--  (707) 

And  omitting  the  terms  involving  the  friction  and  stiffness  of 
cordage, 

P    _  ^ 

^  -    P  ' 

that  is,  the  power  is  to  the  resistance  as  the  radius  of  the  arbor 
is  to  that  of  the  wheel ;  which  relation  is  exactly  the  same  as 
that   of  the   common   lever, 

FIXED   PULLEY. 

|3S1.— The  pulley  is  a  small  wheel  having  a  groove  in  its  cir- 
cumference  for    the   reception    of  a   rope,    to    one    end   of  which    the 


45S 


ELEMENTS     OF    ANALYTICAL    MECHANIC'S. 


power  P  is    applied,  and   to   the  other  the  resistance   Q.     Tlie  pulley- 
may  turn  either  upon   trunnions  or  aoout  an  axle,  supported  in  what 


is  called  a  bloch.  This  is  usually  a  solid  piece  of  wood,  through 
which  is  cut  an  opening  large  enough  to  receive  the  pulley,  and 
allow  it  to  turn  freely  between  its  cheeks.  Sometimes  the  block  is 
a  simple  framework  of  rnctal.  When  the  block  is  stationary,  the 
pulley  is  said  to  be  JixeJ.  The  principle  of  this  machine  is  obvi- 
ously the  same    as  that   of  the  wheel  and  axle. 

The  friction  between  the  rope  and  pulley  will  be  sufficient  to 
give   the    latter    motion. 

Making,  in  Equations  (705)  and  (706), 


0 


P  =  ih 


we    have 


iT  =  4-  ^{w  +  ^  +  P  cos  9)2  +  P2  sin2  (p  =  W  -  ■  (708) 


Making  JR  =z  r,  and  p  =  p',  in  Equation  (704).  and  substituting 
th(i  above  values  of  i^  and  i\"',  we  have,  after  omitting  the  common 
factor  d  s„ 

PE-  QR-f'^^{w+  Q  +  Pcos(^Y+F'^iim^(?^  d,  •     ^^^-R^zO.  -(709) 


APPLICATIONS. 


453 


Solving  this  equation  with  respect  to  P,  we  find  the  value  of 
the  latter  in  terms  of  the  different  sources  of  resistance.  But  this 
direct  process  would  be  tedious ;  and  it  will  be  sufficient  in  all 
cases  of  pz'actice  to  employ  an  approximate  value  for  P  under  the 
}'adical,  obtained  by  lirst  neglecting  the  terms  involving  friction  and 
stiffiiess  of  cordage. 

Thus,  dividing  by  R  and    transposing,  we  find 


P  =  (?  +  /  j^  ^/{vrV<^'^Vy^^^^WVP^^^  9  -I-  «^/ 


K  +  /  q 
•2Ii 


Now  /'  •  -^    is    usually  a    small  fraction ;    an    erroneous    value   as- 

sumed  for  P  under  the  radical,  will  involve  but  a  trifling  error  in 
the  result.  We  may  therefore  write  Q  for  P  in  the  second  mem- 
ber ;  and  neglecting  the  weight  of  the  pulley,  which  is  always  in- 
significant  in    comparison   to    Q,  we   have 


but 


whence, 


^[1  +/'-^V^(l  +cos(p)]  +  cZ, 


1   -j-  cos  (p  =1  2  COS^  -I-  (p  ; 


K-\-  iq 

'2R 


P=  (2(1  -f  2/'^.cosXa,)  +  c;^ 


K+J_ 
2P 


(710) 


^n) 


In  which  9  denotes  the  angle  A  M  P,  which 
is  the  supplement  of  the  angle  A  C  P,  and  de- 
noting this  latter  angle  by  ^,  we  have 


cos  49  =  sin  -^  0  , 
whence 

P=(2(H-2/|psinlO+rf.^'^^' 


K 


2P 


(712) 


If   the    arc    of    the    pulley,    enveloped    by    the   :'ope,    be    180°,    then 
will 

K  +  iq 


p=q{i  +  2/'.^)  +  f.'. 


2P 


(713) 


454 


ELEMENTS     OF    ANALYTICAL    MECHANICS. 


If  the  friction  and  stiffness  of  ;ordage  be  so  small  as  to  justify  their 
omission,  then  will 

P  =-  Q. 

That   is,  the   power   must    be    equal    to  the   resistance,  and    the  only- 
office  of  the  cord  or  rope  is  to  change  the  direction  of  the  power. 


MOVABLE    PULLET. 

§  382. — In  the  fixed  pulley,  the  resultant  action  of  the  power  and 
resistance  is  thrown  upon  the  trunnion  boxes.  If  one  end  of  the 
rope  be  attached  to  a  fixed  hook  A, 
while  the  power  P  is  applied  to  the 
other,  and  the  pulley  is  left  free  to  roll 
along  the  rope,  the  resistance  W  to  be 
overcome  may  be  connected  with  its 
trunnion,  after  the  manner  of  the  figure ; 
the  pulley  is  then  said  to  be  movabfe, 
and  the  relation  between  the  power  and 
resistance  is  still  given  by  Eq.  ('704,) 
in  which  the  principal  resistance  be- 
comes N  -f  N',  and  the  tension  of  the 
rope  between  the  fixed  point  A^  and  the 
tangential  point  H,  becomes   Q. 

Making  in  Equation  (Y04),  i2  =  /,  p  i=  p',  and  W  =z  N-{-N'=2N, 
we  have 

PR-  QP-f'p  W-d^ 
dividing  by  R,  and  transposing 


2R 


2R 


R  =^  0 


(714) 


(715) 


Eliminating  Q  by  means  of  Equation  (708),  and  solving  the  resulting 
equation  with  respect  to  P,  the  value  of  the  power  will  be  known 
in  terms  of  the  resistances.  The  process  may  be  much  abridged  by 
limiting  the  solution  to  au  approximation,  which  will  be  found  sufli- 
clent  in  practice. 


applicatio:n's. 


435 


Neglecting  the  weight  of  the  pulley,  which  is  always  insignificant 
iti  comparison  with  P  or  §,  and  making  Q  =  F,  which  would  be  the 
case  if  A\e  neglect  friction  and  stiffness  of  cordage,  Equation  (V08), 
gives 


and  because 


or. 


iV^=  LpFz^l^  -V/^(1  +  C0S(p); 

1  -i-  COS  cp  =z  2  cos2  ^  (p  =  2  sin2  ^  <), 
W  =2  Q  .sm^d; 


which,  in  Equation  (Vlo),  gives 


W 


2  sin  i  6  ' 


K  +  I. 


W 


V2  sin  M  /iJ/  ' 


2  sin  ^ 


2i? 


(71(5) 


The    quantity    of    work    is    found    by    multiplying    both    members   by 
Rsi^  in   which  s^   is  the   arc  described  at   the  unit's  distance. 

If  the    arc   enveloped    by  the   rope    be    ]  80°,   then  will  t}  ^  =  90°, 
sin  1^  =  1,  and 

K  -\-^I .  W 


P^TF(^+/'.^)4- 


2R 


(71V) 


If   the    friction   and    stiffness  of    cordage    be   neglected,    then    will, 
Equation  (7lG), 

W  =2P  sin  \  ^, 

and  multiplying  by  i?, 

RW  =  P  .2R  .  sin  \  ^  ; 


JL 


but 
A'hence, 


2R%\\\\^  -  AB; 


R  .  W  z=  P  .  AB 


that  is,  the  power  is  to  the  resistance  as  the 
radius  of  the  pulley  is  to  the  cord  of  the  arc 
enveloped  hy  the  rope. 


IV 


456 


5^^ 


ELEMENTS     OF     ANALYTICAL    MECHANICS 


§  383. — The  Mujffle  is  a  collection  of  pulleys  in  two  separate 
blocks  or  frames.  One  of  these  blocks  is  attached  to  a  fixed  point 
A,  by  -which  all  of  its  pulleys  become  fixed, 
while  the  other  block  is  attached  to  the  resist- 
ance  W,  and  its  pulleys  thereby  made  mov- 
able. A  rope  is  attached  at  one  end  to  a  houk 
h  at  the  extremity  of  the  fixed  block,  and  is 
passed  around  one  of  the  movable  pulleys, 
then  about  one  of  the  fixed  pulleys,  and  so  on, 
in  order,  till  the  rope  is  made  to  act  upon  each 
pulley  of  the  combination.  The  power  P  is 
applied  to  the  other  end  of  the  rope,  and  the 
pulleys  are  so  proportioned  that  the  parts  of 
the  rope  between  them,  when  stretched,  are 
parallel.  Now,  suppose  the  power  P  to  main- 
tain in  uniform  motion  the  point  of  applica- 
tion of  the  resistance  W;  denote  the  tension 
of  the  rope  between  the  hook  of  the  fixed 
block  and  the  point  where  it  comes  in  con- 
tact with  the  first  movable  pulley  by  t-^;  the 
radius  of  this  pulley  by  R-^  ;  that  of  its  eye 
by  j-j ;  the  co-efficient  of  friction  on  the  axle 
by  /;  the   constant  and    co-efficient  of   the    stitT- 

ness  of  cordage  by  K  and  /,  as  before ;  then,  denoting  the  tension  of 
the  rope  between  the  last  point  of  contact  with  the  first  movable, 
and  first  point  of  contact  with  the  first  fixed  pulley,  by  4,  the  quan- 
tity of  work   of   the  tension   L^  will,   Equation   (052),   be 

U  Pi  5i  =  ^"1  7?i  s:  +  d^  —2J^  ^1  *"i  +  /'  (^1  +  ^i)  ''i  ^1 1 
in  which 


/  = 


/r+z^' 


dividing  by  s, , 


t,E,  =  f,R,  +  d,  .  -^^  -Ri+f  {h  +  Q  r,. 


Vly) 


APPLICATIONS. 


457 


Again,  denoting  the  tension  of  that  part  of  the  rope  wiiidi  jiusses 
from  the  first  fixed  to  the  second  movable  pulley  l)y  t^ ,  the  radius 
of  the  first  fixed  pulley  by  E.^ ,  and  that  of  its  eye  liy  r. ,  wa  shall, 
in  like  manner,  have 

.t,B,^  t,  R,  +  d,  ^^—  ^.  +  /'  (<■:  +  /.)  'V       .     ( '  I'J) 

And  denoting  the  tensions,  in  order,  by  f^  and  t^ ,  this  last  being 
equal   to  P,  we  shall  have 

t,  B,  =  t,B,  +  d^  ^^^  •  i?3  +  ./■'  (4  -f-  Q  r,.     .     (Tl'.:)) 

PE,=  t,  B,  +  d,  ^,— ^  B,  +  /'  (/,  +  B)  r,.     .     (7- . ) 

SO  that  we  finally  arrive  at  the  power  P,  through  the  tensions  whit-ii 
are  as  yet  unknown.  The  parts  of  the  rope  being  parallel,  and  the 
resistance  W  being  supported  Ijy  their  tensions,  the  latter  may  ol)- 
viously  be  regarded  as  erpial  in  intensity  to  the  components  of  W; 
hence, 

t,  +  k  +  /3  +  4  =   W;      .....      (7-Jii) 

which,  with  the  preceding,  gives  us  five  erpiations  for  the  determi- 
nation of  the  four  tensions  and  power  B.  This  would  involve  a 
tedious  process  of  eliiniuatiun,  whieh  may  be  avoided .  l)y  contenting 
oui'selves  with  an  appi'oxiniation  whieh  is  four^d,  in  practice,  to  be 
sufficiently  accurate. 

If  the   friction   and   stitTness    be    supposed    zero,    for    the    moment, 
Equations  (718)   to  (721)  become 

62  -^^1  —  tj  -^^]  5 

'3  -flj    ^^    '•<  -"2  ) 
M  B'j^    =    /g  A3  , 

BB,  =  UB,; 
from  which    it    is    apparent,  dividing    out    the    radii    7?,,  B.^,  I?^,   •iie., 


458  ELEMENTS     OF     ANALYTICAL    MECHANICS. 

that  t.i  =  ti,  t^  =  i.^,  t^  —  i-i,  F  =  t^;  and  hence,  Equation  {'i'22) 
becomes 

whence, 

W 

the  denominator  4  being  the  Avhole  number  of  pulleys,  movable  and 
fixed.     Had    there    been   n   pulleys,  then  would 

W 
ti  = 

With  this  approximate  value  of  ^i ,  we  resort  to  Equations  (Vl8) 
to  (''21),  and  find  the  values  of  ^.2,  4?  4?  *^c.  Adding  all  these 
tensions  together,  we  shall  find  their  sum  to  be  greater  than  W, 
and  hence  we  infer  each  of  them  to  be  too  large.  If  we  now 
suppose  the  true  tensions  to  be  proportional  to  those  just  found, 
and  whose  sum  is  W^  >  W,  we  may  find  the  true  tension  corre- 
sponding to  any  erroneous  tension,  as  ^1 ,  by  the  following  propor- 
tion, viz. : 

IF 
W,  :    IF::   t,  :  —tr, 

or,  which   is  the  same  thing,  multiply  each  of  the    tensions  found  by 

W 

the    constant  ratio  -r— '   the    product  will    be    the    true    tensions,  very 

if;         ^  '      "^ 

nearly.  The  value  of  ^  thus  found,  substituted  in  Equation  ('''21), 
will   give    that   of  P. 

Exanq^le. — Let  the  radii  i?i ,  Re, ,  Ri  and  R^ ,  be  respectively 
0,20,  0,39,  0,52,  0,65  feet  ;  the  radii  r,  =  r,  —  r^  —  i\  of  the 
eyes  =  0,06  feet ;  the  diameter  of  the  rope,  which  is  white  and 
dry,  0,79  inches,  of  which  the  constant  and  co-efficient  of  rigidity 
are,  respectively,  K  =  1,6097  and  I—  0,0319501;  and  suppose  the 
pulley  of  l>rass,  and  its  axle  of  wrought  iron,  of  which  the  co-efficient 
/  =  0,09,  and    the   resistance    W  a  weight  of  2400  pounds. 

AVithout   friction    and    stiffiiess  of  cordage, 

2400        -^ '*''• 
t,  =  —-  =  600. 


APPLICATIONS. 


459 


Dividing  Equation  (718)  by  7?,,  it   becomes,  since  d^  =  1, 
A^  +  If, 


f^  + 


2i?. 


Substituting  the  value  of  H, ,  and  the  above  value  of  /, ,  and  regard, 
ing    in    the    last    term    L  as    equal    to  /j ,  which  we   may  do,  because 

of  the    small  co-efficient  -j^  /',   we  find 

600 

1,0097  +  0,0319501   x  GOO 


-  J  + 


2  X  (0,26) 


y  =  623,39. 


+  ^  X  0,09  X  (GOO  4-  GOO) 


Again,  dividing    Equation    C^IO)    by  7?.,,  and    substituting    tliis    value 
of  L,  and   that  of  H^,  we  find 

lbs. 

t^  z^  673,59. 

Dividing  Equation  (720)  by  i?3,  and  substituting  this  value  (f  t.^,  as 
well   as    that   of-Ks,  there  will    result 


lbs. 

t^  =  709,82 ; 


whence, 


TFi  =  t,  -^  t,  +  t,  +  h 


r     600     ^ 

-f  G28,39 
+  673,59 

[  +  T09,82  J 


=  261]  ^'> 


and 


W  2400 


=  0,919 ; 


2611,80 

which  will  give  for  the   true   values   of 

^1  .=:  0,919  X  600       =  551,400 

t,  =  0,919  X  628,39  ^  577,490 

f,  =  0,919  X  673,59  =  619,029 

/,  =  0,919  X  709,82  =  652,324 


2400,243 


460 


ELEMENTS    OF     ANALYTICAL     MECHANICS. 


The  above  value  for  t^  —  052,324,  in   Equation  ('721),  will  give,  after 
dividing  by  R^,  and  substituting  its  numerical  value, 


P  =  \ 


+ 


652,324 

1,6007  +  0,03195  X  652,324 


2  X  0,65 


0.06 


-f  ^  X  0,09  X  (652,324  +  P)  ■ 
U,6o 


md  making  in  the  last  factor  P  =  t^  =  652,324,  we  find 

lbs.  lbs.  Ib^.  lbs, 

P  =  652,324  +  17,270  +  10,831  =  680,425. 

Thus,  without  friction  or  stiffness  of  cordage,  the  intensity  of  P  would 
be  600  lbs. ;  with  both  of  these  causes  of  resistance,  which  cannot  be 
avoided  in  practice,  it  becomes  680,425  lbs.,  making  a  difference  of 
80,425  lbs.,  or  nearly  one-seventh ;  and  as  the  quantity  of  work  of 
the  power  is  proportional  to  its  intensity,  we  see  that  to  overcome 
fi-iction  and  stiffness  of  rope,  in  the  example  before  us.  the  mot(jr 
must  expend  nearly  a  seventh  more  work  ihan  if  these  sources  of 
resistance  did  not  exist. 


THE    WEDGE. 

§  384. — The  wedge  is  usually  employed  in  the  operation  of  cut- 
ting, splitting,  or  separating.  It  consists 
of  an  acute  right  triangular  prism  ABC. 
The  acute  dihedral  angle  A  Cb  is  called 
the  edffe;  the  opposite  plane  face  Ab 
ihe  back;  and  the  planes  Ac  and  Cb, 
wiiich  terminate  in  the  edge,  the  faces. 
The  more  common  application  of  the 
wedge  consists  in  driving  it,  by  a  blow 
upon  its  back,  into  any  substance  which 
we  wish  to  split  or  divide  into  parts,  in 
such  manner  that  after  each  advance  it 
shall  be  supported  against  the  faces  of 
the    opening    till    the    work    is    accomplished. 


APPLICATIONS. 


4G1 


§  aSo. — The  blow  by  wliieli  tlie  wedge  is  driven  forM<trd  will  be 
supposed  perpeiKliculiir  to  its  back,  for  if  it  were  oblitjue,  it  would 
only  tend  to  impart  a  rotary  motion,  and  give  rise  to  complications 
which  it  would  be  unprofitable  to  consider :  and  to  make  the  case 
conform  still  further  to  practice,  we  will  suppose  the  wedge  to  be 
isosceles. 

The  wedge  ACB  being  inserted  in  the  opening  «///;.  and  in  eon- 
tact  with  its  jaws  at  a  and  6,  we  know 
that  the  resistance  of  the  latter  will 
be  pei'pendicular  to  the  faces  of  the 
wedge.  Through  the  points  a  and  b 
draw  the  lines  uq  and  h -p  normal  to 
the  faces  A  C  and  B  C ;  from  their 
point  of  intersection  0  lay  oif  the 
distances  Oq  and  0 j)  equal,  respec- 
tively, to  the  resistances  at  a  and  h. 
Denote  the  first  by  Q^  and  the  second 
by  P.  Completing  the  pjarallelognim 
0  qinp^  Om  will  represent  the  re- 
sultant of  the  resistances  Q  and  F. 
Denote  this  resultant  by  R\  and  the 
angle  A  C B  of  the  wedge  by  ^,  which, 
in    the    quadrilateral    a  0  b  C^   will    be 

equal  to  the  supplement  of  the  angle  a  Ob  ^  -pOq.  the  angle  ninde 
by  the  directions  of  Q  and  P.  From  the  parallelogram  cf  forees, 
\\Q  have, 

R' "^  =  P-+  Q^^  +  2P  Q  cosp  Oq  =  P^-{-  Q-  -2P  Q  cos  0  ; 


or, 


P' 


^/pz  -\-  Q2  -  2  P  Q  cos  6. 


The  resistance  Q  will  produce  a  friction  on  the  face  A  C  equal 
to  fQ,  and  the  resistance  P  will  produce  on  the  face  i?  C  the  fric- 
tion /  P :  these  act  in  the  directions  of  the  fiices  of  the  wedge. 
Produce  them  till  they  meet  in  C.  and  lay  oil'  the  ibstanccs  C q'  and 
Op'  to    rejjresent    theij-    inli-nsities,    and     complete    the    ]iaralle!ogram 


462  ELEMENTS     OF     ANALYTICAL     MECHANICS. 

Cq'  0'  2>'  '■,    CO'  will  represent  the  resultant  of  the  frictions.     r><^note 
this   by   11".  and  we  have,  from  the  parallelogram  of  forces, 

R'"^  =  P  Q^  +  /2  p2  +  2/2  F  Qcos6; 


R"  =  f  yp2  +  Q2  +  2  P  Q^  cos  6. 

The  wedge  behig  isosceles,  the  resistances  jP  and  Q  will  be  equal, 
their  directions  .being  normal  to  the  faces  will  intersect  on  the  line 
CD,  which  bisects  the  angle  C  =  &,  and  their  resultant  will  coin- 
cide with  this  line.  In  like  manner  the  frictions  will  be  equal,  and 
their  resultant  will  coincide  with  the  same  line.  INIaking  Q  and  P 
equal,  we  have,  from  the  above  equations, 


R'  ==     P  V-i^  -  cosO), 

R"  z=:  fP   V2  (1    +    cost)). 

But, 

1  —  cos  ^  =  2  sin2  1  S, 

1  +  cos  ^  =  2  cos2  1  6  ; 
whence  we  obtain,  by   substituting  and  reducing, 

R'   =  2  P.  sin  i  1 

R"  =  2/.  P.  cos  1  6  ; 
and  farther, 


therefore. 


sin 

\^ 

= 

2  A  6" 

cos 

h^ 

— 

CD 
A  C  ' 

R' 

= 

P 

AB 

A  C" 

R" 

c 

J/' 

p.  ^^ 
^  A&- 

Denote  by  F  the   intensity  of  the  blow  on   the    back  of  the  wedge. 
If  this   blow  be  just    sufficient    to  produce   an   equilibrium    bordering 


APPLICATIONS. 


463 


on  motion  forward,  call  it  F' ;  the  friction  will  oj^pose  it,  and  we 
must   have, 

i^'==i2'+i2"  =  P.^+2/.P.^^-     .     .     .     (723) 

If,  on  the  contrary,  the  blow  be  just  sufficient  tc  prever.t  the  wedge 
from  flying  back,  call  it  F" ;  the  friction  will  aid  it,  and  we  must 
have, 

AC  ''AC  ^       ' 

The  wedge  will  not  move  under  the  action  of  any  force  whose  Inten- 
sity is  between  F'  and  F".  Any  force  less  than  F",  will  allow  it 
to  fly  back;  any  force  greater  than  F',  will  drive  it  forward.  Tlic 
range  through  which  the  force  may  vary  without  producing  motion, 
is  obviously, 

F'  -  F"  =  ^fP-^-^ (725) 

AC  ^ 

which  becomes  greater  and  greater,  in  proportion  as  (7i>  and  A  G 
become  more  nearly  equal;  that  is  to  say,  in  proportion  as  the 
wedges  becomes  more  and  more  acute. 

The  ordinary  mode  of  emj^loying  the  wedge  requires  that  it  shall 
retain  of  itself  whatever  position  it  may  be  driven  to.  This  makes 
it  necessary  that  F"  should  be  zero  or  negative,  Eq.  ('J'24),  whence 

^     A  C   ~     ''  A  C  A  C    ^     •'  AC   ' 

or,  omitting  the  common  foctors  and  dividing  both  members  of  the 
equation    and    inequality  by  2  (7  Z>, 


CD 

^  A  B 

but     ^  is    the  tangent  of  the    angle  A  C  D ;  hence  wc  conclude, 

that  the  wedge  will  retain  ^s  place  when  its  semi-angle  does  not 
excGHid  that  whose  tangent  is  the  co-efficient  of  fricticn  between  the 
surface  of  the  wedge  and  the  surface  of  the  opening  which  it  is 
intended  to  enlarge. 


404 


ELEMENTS     OF     ANALYTICAL     MECHANICS. 


Resuming  Eq.  (724),  and  snpposing  the  last  term  of  the  second 
member  greater  than  the  first  term,  F"  becomes  negative,  and  "will 
represent  the  intensity  of  thi;  force  necessary  to  withdraw  the  wedge ; 
which  will  obviously  be  the  greatest  possible  when  A  B  is  the  least 
possible.  This  explains  why  it  is  that  nails  retain  Avith  such  perti 
nacity  their  i)laces  when  driven  into  wood,  &c. 


n 


THE    SCKEW. 


§  38G. — The  Screw,  regarded  as  a  mechanical  power,  is  a  device  by 
which  the  principles  of  the  inclined  plane  are  so  applied  as  to  pro- 
duce considerable  pressures  with  great  steadiness  and  regularity  of 
motion. 

To  form  an  idea  <jf  the  figui-c  of  a  screw  and  its  mode  of  acti(jii. 
coticeive  a  right  cylinder,  a  /.-,  with  circular  base,  and  a  rectangle,  or 
other  plane  figure,  abcm,  having  one  of  its  sides 
ah  coincident  with  a  surface  element,  while-  its 
plane  passes  through  the  axis  of  this  cylinder. 
Next,  suppose  the  plane  of  the  generatrix  to 
rotate  uniformly  about  the  axis,  and  the  gener- 
atrix itself  to  move  also  uniformly  in  the  direc- 
tion of  that  line  ;  and  let  this  twofold  motion 
of  rotation  and  of  translation  be  so  regulated, 
that  in  one  entire  revolution  of  the  plane,  the 
generatrix    shall    progress    in    the    direction    of 

the  axis  over  a  distance  greater  than  the  side  ab,  which  is  in  the 
surface  of  the  cylinder.  The  generatrix  will  thus  generate  a  pro- 
jecting and  Avinding  solid  called  a  Ji//cf,  leaving  between  its  turns 
a  groove  called  the  channel.  Each  point  as  m  in  the  perimeter 
of  the  generatrix,  will  generate  a  curve  called  a  helix,  and  it  is 
obvious,  from  what  has  been  said,  that  every  helix  will  enjoy  this 
property,  viz.  :  any  one  of  its  points  as  m,  being  taken  as  an  origin 
of  reference,  as  well  for  the  curve  itself  as  f)i-  its  projection  on  a 
plane  through  this  point  and  at  right  angles  to  the  axis,  the  distances 
d'  m\  d"  m",  &ic.,  of  the  several    points  of  the  helix  from    this  plane, 


APT  Ligations. 


465 


are  respectively  proportioned  to  the  circular  arcs  vid\  md",  &c., 
into  which  the  portions  mm',  mm",  &c.,  of  the  helix,  between  the 
origin    and    these    points,    are    projected. 

The  solid  cylinder  about  which  the  fillet  is  wound,  is  called 
the  neivel  of  the  screw ;  the  distance  m  m'",  between  the  consecu- 
tive turns  of  the  same  helix,  estimated  in  the  direction  of  the  axis, 
is    called    the   helical   interval. 

The  fillet  is  often  generated  by  the  motion  of  a  triangle  with 
one  of  its  sides  coincident  with  a  b  ;  and  as  the  discussion  will  be 
more  general  by  considering  this  mode  of  generation,  we  shall  adopt 
it.  The  surfaces  of  the  fillet,  which  are  generated  by  the  inclined 
fiices  of  the  triangle,  are  each  made  up  of  an  infinite  number  of 
helices,  all  of  which  have  the  same  interval,  though  the  helices 
themselves  are  at  different  distances  from  the  axis,  and  have  different 
inclinations   to   that  line. 

The  inclination  of  the  different  helices  to  the  axis  of  the  screw 
increases  from  the  newel  to  the  exterior  surflxce  of  the  fillet, 
the  same  helix  preserving  its 
inclination  unchanged  throughout. 
The  screw  is  received  into  a  hole 
in  a  sylid  piece  B  of  metal  or 
wood,  called  a  7iut  or  burr.  The 
surfiice  of  the  hole  through  the 
nut  is  furnished  with  a  winding 
fillet  of  the  same  shape  and  size 
as  the  channel  of  the  screw,  so 
that  the  surfaces  of  the  screw  and 
nut  are  brought  into  accurate  con- 
tact. 

From  this  arrangement  it  is 
obvious  that  when  the  nut  is  sta- 
tionary, and  a  rotary  motion  is 
communicated  to  the  screw,  the 
latter  will  move  in  the  direction 
of    its    axis ;     also,    when    the    screw    is    stationary    and    the   nut    is 

turned,    the   nut    must   also    move   in   the   direction    of  the   axis.      In 

30 


JX- 


T. 


4:Q6 


ELEMENTS  OF  ANALYTICAL  MECHANICS. 


the  first  case,  one  entire  revolution  of  the  screw  will  carry  it  lon- 
gitudinally through  a  distance  equal  to  the  helical  interval,  and  any 
fractional  portion  of  an  entire  revolution  will  carry  it  through  a  pro- 
portional distance ;  the  same  of  the  nut,  vrhen  the  latter  is  mova- 
ble and  the  screw  stationary.  The  resistance  Q  is  applied  either  to 
the  head  of  the  screw,  or  to  the  nut,  depending  upon  which  is  the 
movable  element ;  in  either  case  it  acts  in  the  direcliou  D  C  of 
the  axis.  The  power  F  is  applied  at  the  extremity  of  a  bar  G  H 
connected  with  the  screw  or  nut,  and  acts  in  a  plane  at  right 
angles  to   the    axis   of  the    screw. 

From    the    description    of  the    screw  and    its   mode    of  generation, 
we    may  find    the    equation    of   its    fillet    or    helicoidal    surface.      For 
this  purpose,  take   the  axis  z  to   coincide  with  the   axis  of  the  newel, 
and    the    initial    position  of  the   generatrix  in    the    plane  y  z.     Make 
A-   =  any    definite    portion    C  C 

of  an  assumed  helix  ; 
(p  =  the    angle    Y  A  t^    through 
which    the    rotating    plane 
has  turned  during  the  gene- 
ration of  s ; 
r  —  the    distance    CD    of    this 

helix  from    the   axis  z ; 
a  =  the    angle  which  this    helix 
makes  with  the  plane  x  y ; 
§  =  the  angle   C BD  which  the 
generatrix  of  the  helicoidal 
surface     makes     with     the 
axis  z ; 
y  —  the  co-ordinate  AB  oi  the 
point  in  which   the  genera- 
trix, in  its  initial    position,  intersects    the   axis  z. 
Then,  for   any  point   as   C  of  the  generatrix  in  its  initial  position, 
we   have 

z=.AD  =  AB  +  BD  =  j-\-r.  cotan  ?, 
and    for   any  subsequent  position,  as   C  B'. 

z  —  Y  -\-  r  .  cotan  §  -f  ?• .  9  .  tan  a,  •     .     •     .     (726) 


(^-t   - 


V  .U^  &.  (*-  y 


APPLICATIONS.  467 

which  is  the  equation  sought,  and  in  which  a  and  r  are  •jonstant 
for   the   same   helix,  and  variable  from  one  helix    to  another 

The   power   P   acts  in    a   direction    perpendicular    to   the    axis  of 
the  newel.     Denote   by  I  its   lever  arm  ;  its  virtual    moment  will  be 

Pld;^. 

The  resistance  Q  acts  in  the  direction  of  the  axis  of  the  newel ; 
its    virtual    moment  will    be 

Qdz. 

The  friction  acts  in  the  dii'ection  of  the  helicoidal  surface  and  j^aral- 
lel  to  the  helices.  Conceive  it  to  be  concentrated  upon  a  mean 
helix,  of  which  the  distance  from  the  newel  axis  is  ?•,  and  length  *• : 
denote  the  normal  pressure  by  N,  and  co-efficient  of  friction  by  /. 
The  virtual  moment  of  friction  will  be 

f.N.cU- 

and  Equation  (645), 

Pld(^—Qdz—f.N.ds^Q (727) 

But  the  displacement  must  satisfy  Equation  (726),  or,  as  in  §  213, 
the  condition, 

Z  =  2  —  ?'  .  9  .  tan  a.  —  r  .  cotan  §  —  y  :=  0 ;       .     (728) 

and  also, 

r  =  constant C^^O) 

Differentiating,  we  have, 

dz  —  cotan  ^  .  d  r  —  r  tan  a  </<?  =  0, 

dr  -  Q.  1 

Multiplying  the  first  by  X,  the  second  by  X',  adding  to  Eqiiation 
(727),  and  eliminating  ds  by  the  relation 

d  s  zzz  r  .  do  .  cos  a  +  c7  2  .  sin  a,    .  .  (730) 

we  find, 
{P I  —f . N .cos, a .r  —  \isixa.r)d<f  +  {\  -  Q-  f.Ksma) d z  Jr  {\'  —  Xcotaug)Q'r=  0 


46S         ELEMENTS     OF     ANALYTICAL     MECHANICS, 
and,  from  the  principle  of  indeterminate  co-efficients, 

PI  —  f .  iV  .  cos  a  .  r  —  X  .  tan  a  .  r  =  0  ;      .      .      (7-T 1 ) 

Q  +fN.  sin  a  -  X  =  0; (7.3_') 

X'  —  X  cotan  g  =  0 (732)' 

The    variables    d  z^  d  r,  andrtZtp,  are    rectangular;     whence,    Equation 
(331), 


*  =  H/  (-,77)'+  (■— )  +  ^7^)'=  ^  vT"^rS^«  +  c„tm,=  S. 

Substituting  this  in  Equations  (73 1)  and  (732),  and  eliminating  X. 
there  will  result 


r      tan  a   -f-  /.  cos  a  .  -y/l  +  tan^  a  -f-  cotan^b 
1  — /.  sin  a.  .  -^/l  +  tan'"  a  +  cotan^§ 


■/        uiii  u.    -f-  /  .  (JUS  a  .  v  1    "+-   laii-  a  -f-  coian'-b 


Substituting    the    value    of   X    from    Equation    ('732),  in    Equation 
(732)',   we  find, 

-V  /         ^  cotan  § 

>^'  =  (2 7-^ ; ,=^^  ;    .       73-t) 

1  — /.  sui  a  -/I  -t-  tan2  a  -f  cotan^  § 

in  which  X'  is,  §  217,  the  value    of  the    force    acting    in    the  direction 
of  r. 

§  387.— If  the  fillet    be    rectangular,  §  —  90°,  cotan  §  =  0,  and 


r     tan  «  +  /•  cos  «  .  ^1  +  tan^  « 
^  1  —  / .  sin   a  .  ^i  +  tan2  a 

and 

X'  r=  0. 

§  388. — If  -we  neglect  the  friction,  /  =  0  ;  and 

F  i  =  Q  .  r  .  tan  a, 

multiplying  both  members  by  2  *, 

F  .2':r  I  =z  (2.2*r.  tan  k (730) 

That    is,   Me  power  is  to  the  resistance   as    t/te  helical   interval   is    to 
the  circumference  described  by  the  end  of  the  lever  arin  of  the  power. 


APPLICATIONS. 


469 


puMrs. 


§  389. — Any  machine  used  for  raising  liquids  from  one  level 
to  a  higher,  in  which  the  agency  of  atmospheric  pressure  is  employed, 
is  called  a  Pump.  There  arc  various  kinds  of  pumps ;  the  more 
common  are  the  sucking,  forcing,  and  lifllng  pumij)s. 

§  300. — The  SacVing-Pump  consists  of  a  cylindrical  body  or  barrel 
7j,  frum  the  lower  end  of  which  a  tube  Z>,  called  the  sucking-pipe, 
descends  into  the  water  contained  in  a  reservoir  or  well.  In  the 
interior  of  the  l)arrel  is  a  movable  piston  C,  surrounded  v.ith  leather 
to  make  it  water-tight,  yet  ca- 
pni.le  of  moving  up  and  down 
freely.  The  piston  is  j)erforated 
in  the  direction  of  the  bore  of 
the  barrel,  and  the  orifice  is 
covered  by  a  valve  F  called 
the  lihtoii-valve^  which  open.s  up- 
ward ;  a  simihir  valve  E^  failed 
the  sleeping-valve^  at  tlie  bottom 
of  the  barrel,  covers  the  upper 
end  of  the  sucking-pipe.  Above 
the  highest  point  ever  occupied 
by  the  piston,  a  discharge-pipe 
P  is    inserted    into    the    barrel  ; 

the  piston    is  worked    by  mi'ans  _^  _^ 

of  a  lever  //,  or  other    contriv-  'm:^ 

anee,  attached    to  the  piston-rod 

O.  The  distanced  J',  between  the  highest  and  lowest  points  of  the 
piston,  is  called  the  ^^foy.  To  explain  the  action  of  this  pump,  let 
the  piston  be  at  its  lowest  point  yl,  the  valves  E  and  /''  closed  by 
their  own  weight,  and  the  air  within  the  pump  of  the  same  density 
and  elastic  force  as  that  on  the  exterior.  The  water  of  the  reservoir 
will  stand  at  the  same  level  L  L  both  within  and  without  the 
sucking-pipe.  Now  suppose  the  piston  raised  to  its  highest  point  .1', 
the    air    contained    in    the    barrel     and    sucking-pipe    will    tend    by    its 


470  ELEMENTS     OF     ANALYTICAL     MECHANICS. 

elastic  force  lo  occupy  the  space  which  the  piston  leaves  void,  the 
valve  E  will,  therefore,  be  forced  open,  and  air  will  pass  from  the 
pipe  to  the  barrel,  its  elasticity  diminishing  in  propoition  as  it  fills 
a  larger  space.  It  will,  therefore,  exert  a  less  ])ressure  on  the 
water  below  it  in  the  sucking-pipe  than  the  exterior  air  does  on  that 
in  the  reservoir,  and  the  excess  of  pressure  on  the  part  of  the 
exterior  air,  will  force  the  w-ater  up  the  j^ipe  till  the  weight  of  the 
suspended  column,  increased  by  the  elastic  force  of  the  internal  air, 
becomes  equal  to  the  pressure  of  the  exterior  air.  When  this  talces 
place,  the  valve  E  will  close  of  its  own  weight ;  and  if  the  piston 
be  depressed,  the  air  contained  between  it  and  this  valve,  having 
its  density  augmented  as  the  piston  is  lovvered,  will  at  length  have 
its  elasticity  greater  than  that  of  the  exterior  air ;  this  excess  of 
elasticity  will  force  open  the  valve  F,  and  air  enough  will  escape 
to  reduce  v.'hat  is  left  to  the  same  density  as  that  of  the  exterior 
air.  The  valve  F  will  then  fall  of  its  own  weight ;  and  if  the 
piston  be  again  elevated,  the  w^ater  will  rise  still  higher,  for  the 
same  reason  as  before.  This  oj)eration  of  raising  and  depi'essing 
the  piston  being  repeated  a  few  times,  the  water  wnll  at  length  entei 
the  barrel,  through  the  valve  F^  and  be  delivered  from  the  dis- 
cliarge-pipe  P.  The  valves  F  and  F^  closing  after  the  water  has 
passed  them,  the  latter  is  prevented  from  returning,  and  a  cylinder 
of  water  equal  to  that  through  which  the  piston  is  raised,  will,  at 
each  >q)ward  motion,  be  forced  out,  provided  the  discharge-})ipe  is 
large  enough.  As  the  ascent  of  the  water  to  the  piston  is  pro- 
duced by  the  difference  of  pressure  of  the  internal  and  external  air, 
it  is  plain  that  the  lowest  point  to  which  the  piston  may  reach, 
should  never  have  a  greater  altitude  above  the  water  in  the  reser 
voir  than  that  of  the  column  of  this  fluid  which  the  atmospheric 
pressure    may    suppcM't,    ir    vacuo,    at    the    place. 

§391. — It  will  readily  appear  that  the  ri-e  of  water,  during 
each  ascent  of  the  piston  after  the  first,  depends  upon  the  expuh-^iou 
of  air  through  the  ])iston-valve  in  its  ]irevious  descent.  But  air  can 
onlv  issue  through  this  valve  when  thi-  ;;ir  helow  it  has  a  greater 
density    and    therelbre    greater    elasticity    thai,    the    external    air  ;    and 


APPLICATIONS. 


471 


if  the    piston    may   not   descend    low    enough,    fur    want   of  sufficient 
play,  to   produce    this  degree  of  compression,   the  water   must   cease 
to  rise,  and  the  working  of  the  piston  can  have  no  other  eflect  than 
alternately  to    compress    and    dilate    the    same 
air    between    it    and    the  surface   of   the  water. 
To    ascertain,  therefore,  the    relation  which   the 
play  of  the   piston    should    bear    to   the    other 
dimensions,  in    order    to  make  the  pump  etfec- 
tive,  suppose  the  water   to  have  reached  a  sta- 
tionary level   X,    at    some    one    ascent    of   the 
piston  to  its  highest   point  A\  and    that,  in  its 
subsequent    descent,    the    piston-valve    will    not 
open,  but   the  air  below  it  Avill    be  compressed 
only  to  the  same  density  with   the  external  air 
when    the  piston    reaches    its    lowest   point  ^-1. 
The    piston    may  be  worked    up  and   down    in- 
definitely,   within     these    limits    fur     the    play,  ~-^g^^"" 
without    moving    the  water.      Denote    the   play 

of  the  piston  by  a ;  the  greatest  height  to  which  the  [>iston  may  be 
raised  above  the  level  of  the  water  in  the  reservoir,  by  i,  which  may 
also  be  regarded  as  the  altitude  of  the  discharge  pipe;  the  elevation 
of  the  point  X,  at  which  the  water  stops,  above  the  water  in  the 
reservoir,  by  x ;  the  cross-section  of  the  interior  of  the  barrel  by  B. 
The  volume  of  the  air  between  the  level  A"  and  A  will  be 


B  X  {h 


«) 


the  volume  of  this  same  air,  wlien  the  piston  is  raised  to  A\  pro- 
vided the  water   does  not   move,  will    be 

B{h-.r). 

Eepresent  by  h  the  greatest  height  to  \\\\k\\  water  may  be  supported 
in  vacuo  at  the  place.  The  weight  of  the  column  of  water  which 
the  elastic  force  of  the  air,  when  occupying  the  space  between  the 
limits  X  and  A,  will  supp.rt  in  a  tube,  with  a  bore  equal  to  that 
of  tlie  barrel    is   measure  1    l)y 

Bh.g  .  D; 


473  ELEMENTS     OF    ANALYTICxVL    MECHxVN.'^S. 

hi  which  D  is  the  density  of  the  water,  and  (j  the  ll  rcu  of  gravity. 
The  Aveight  of  the  column  which  the  elastic  force  of  ih  .3  same  air 
will    support,  when    expanded    between    the    limits  X  and  A\   will  be 

Bh'  .(J  .D; 

in  Mhich  W  denotes  the  height  of  this  new  column.  But,  fr. m  Ma- 
riotte's    law,  we  have 

B{}j  -  X  -  a)   :  B{b  -  x)   :  :  B  h'  g  B  :  B  h  rj  D  ; 
wlKiice, 

6  —  X 

But  there  is  an  equilibrium  between  the  pressure  of  the  external 
air  and  that  of  the  rarefied  air  between  the  limits  X  and  A',  when 
the  latter  is  increased  by  the  weight  of  the  column  of  water  whose 
altitude  is  x.     Whence,  omitting  the  comnnjn  factors  i?,  D  and  g, 

h  —  X  —  a 

X  -f  h'  =  X  +  h =  h  ■ 

b  —  X 

or,  clearing  the  fraction  and  solving  the  equation  in  reference  to  x, 
we  find 


X  =  ^h  ±\y'b'  -  4:ah. (737) 

When  X  has  a  real  value,  the  water  will  cease  to  rise,  but  x 
will  be  real  as  long  as  i-  is  greater  than  4  a  h.  \^^  on  the  con- 
trary, 4«A  is  greater  than  P,  the  value  of  x  will  be  imaginary,  and 
the  water  cannot  cease  to  rise,  and  the  pump  will  always  be  effective 
when    its    dimensions    satisfy  this   condition,  viz.  : — 

4  a  A  >  h"^, 
or, 

">4l' 

that  is  to  say,  the  j)l(ig  of  the  piston  vmst  he  greater  than  the  square 
of  the  altitude  of  the  upper  limit  of  the  plaij  of  the  piston  above 
the  surface  of  the  water  in  the  reservoir^  divided  by  four  times  the 
height   to  which   the  atmospheric  pressure  at  the  place^  where   tlie  pump 


APPLICATIONS. 


473 


is  used,  will  support  water  in  vacuo.  This  last  height  is  easily  found 
by  means  of  the  barometer.  We  have  but  to  notice  the  altitude 
of  the  barometer  at  the  place,  and  multiply  its  column,  reduced  to 
feet,  by  13|,  this  being  the  siiecific  gravity  of  mercury  referred  to 
water  as  a  standard,  and  the  product  will  give  the  value  of  k  in 
feet. 

£xam2}le. — Required  the  least  play  of  the  piston  in  a  sucking- 
pump  intended  to  raise  water  through  a  height  of  13  feet,  at  a 
place  Avhere    the    barometer   stands    at    28    inches. 


Iler. 


b  =  VS,     and     P  =  109. 


Barometer, 


Play 


28 

—  =z  2,333  feet. 


h 


ft- 


2,33a  X   13,5 
109 


IP 
^  4h         4  X  31,5 


31.5  feet. 


=  1,341  +  ; 


that    is,  the    play  of  the    piston    must   be  greater    than    one    and    one 
third  of  a    foot. 

§  392. — The  quantity  of  work  perfc^nned  by 
the  motor  during  the  deliv(!ry  of  water  through 
the  discharge-pipe,  is  easily  computed.  Sup- 
pose the  piston  to  have  any  position,  as  M, 
and  to  be  moving  upward,  the  water  being 
at  the  level  LL  in  the  reservoir,  and  at  P 
in  the  pump.  The  pressure  upon  the  upper 
surface  of  the  piston  will  be  equal  to  the 
entire  atmospheric  pressure  denoted  by  A, 
increased  by  the  weight  of  the  column  of 
water  IIP',  whose  height  is  c',  and  whose 
base  is  the  area  B  of  the  piston  ;  that  is,  the  • 
pressure   upon    the    top  of  the   piston  will  be 

A  +  jBc'gD, 

in  which  r/  and  D  a^e  the  force  of  gravity  and  density  of  the  water, 
respectively.      Again,    the   pressure    upon    the    under    surface    of    the 


It 
X  JV 

z  c' 

474  ELEilENTS     OF     ANALYTICAL    MECHANICS. 

piston  is  equal  to  the  atmosplieric  pressure  J,  transmitted  through 
the  water  in  the  reservoir  and  up  the  suspended  column,  diminished 
by  the  weight  of  the  column  of  water  N 31  below  the  piston,  and 
of  which  the  base  is  B  and  altitude  c ;  that  is,  the  pressure  from 
below  will    be 

A  -  BcgD, 

and    the    difference  of  these   pressures  AviU    be 

A  +  Be'  (jD  -  {A  -  Beg  D)  =  BgD{c  +  e')  ; 

but,  employing  the   notation  .of  the    sucking-pump   just   described, 

c  +  c'  =  i  ; 

whence,  the  foregoing   expression    becomes 

Bb.g.I); 

which  is  obviously  the  weight  of  a  column  of  the  fluid  whose  base 
is  the  area  of  the  piston  and  altitude  the  height  of  the  discharge-pipe 
above  the  level  of  the  water  in  the  reservoir.  And  adding  to  this 
the  effort  necessary  to  overcome  the  friction  of  the  parts  of  the  pump 
when  in  motion,  denoted  by  9,  we  shall  have  the  resistance  which  the 
force  F,  applied  to  the  piston-rod,  must  overcome  to  produce  any 
useful  effect ;  that  is, 

F  -  B  fj  g  B  +  cp. 

Denote  the  play  of  the  piston  by  ji;,  and  the  r.umber  of  its  double 
strokes,  from  the  beginning  of  the  flow  through  the  discharge-pipe 
till  any  quantity  Q  is  delivered,  by  « ;  the  quantity  of  work  will,  by 
omitting  the  effort  necessary  to  depress  the  piston,  be 

F712J  =  np[Bb  .gB  +  cp]; 

or  cstinuiting  the  volume  in  cubic  feet,  in  which  case  ;;  and  b  must 
be  expressed  in  linear  feet  and  B  in  square  feet,  and  substituting  for 
g  D  its  value  62,5  pounds,  we  finally  have  for  the  quantity  of  work 
necessary  to  deliver  a  number  of  cubic  feet  of  water   Q  =  Bn])^ 

Fii])  —  np  [()2,5  .  Bb  +  cp];      .     .     .     .     (738) 

in    which    cp    must    be   expressed    in    pounds,  and    may   be  determined 


APPLICATIONS. 


4,5 


either  by   experiment  in   each   })articular   pump,  or   computed    hy  the 
rules  already  given. 

It  is  apparent  that  the  action  of  the  sucking-pump  must  lie  very 
irregular,  and  that  it  is  only  during  the  ascent  of  tlie  piston  that  it 
produces  any  useful  effect ;  during  the  descent  of  the  piston,  the  force 
is  scarcely  exerted  at  all,  not  more  than  is  necessary  to  overcome 
the  friction. 

§  393. — The  Lifting-Pumii  does  not  differ  much  from   the  sucking- 
pump  just  described,  except  that  the   barrel   and  sleeping-valve  E  are 
placed  at  the  bottom  of  the  pipe,   and    some   distance   below    the  sur- 
face of  the  water  L  L  in  the  reservoir  ;  the 
piston    may    or    may    not    be    below     this 
s;inie   surface  when  at  the   lowest  point  of 
its   jilay.       The    piston    and    sleeping-valves 
open   upAvard.     Supposing  the  piston  at  its 
lowest  point,  it  will,  when   raised,  lift  the 
colunm   of   water    above    it,   and    the  pres- 
sure of  the   external   air,  together  with  the 
head   of    fluid   in    the    reservoir    above    the 
level   of   the   sleeping-valve,   will   force   the 
latter  open  ;    the   water  will    flow    into    the 
barrel  and  follow    the    piston.       When    the  :  % 

piston   reaches   the   upper  limit  of  its  play, 
the    sleeping-valve    will    close    and    prevent  =sq=pia^-- 

thc    retui-n    of    the    water    above    it.       The 

j^istou  being  depressed,  its  valves  F  will  open  and  the  Avater  will 
flow  thi'ough  them  till  the  piston  reaches  its  lowest  point.  The 
same  operation  being  repeated  a  few  times,  a  colunm  of  water  will 
be  lifted  to  the  mouth  of  the  discharge-pipe  P,  after  which  every 
elevation  of  the  piston  will  deliver  a  volume  of  the  fluid  equal  to 
that  of  a  cylinder  whose  base  is  the  area  of  the  piston  and  whose 
altitude  is  equal  to  its  play. 

As  the  water  on  the  same  level  Avithin  and  without  the  pump 
will  be  in  equilibrio,  it  is  plain  thot  the  resistance  to  be  overcome 
hy  the  power  will  hi    the  friction  of  the  rubbing  surfaces  of  the  pumpj 


rv 


476 


ELEMENTS     OF     ANALYTICAL    MECHANICS. 


augmented  by  the  weight  of  a  column  of  fluid  whose  base  is  the  area 
of  the  piston,  and  altitude  the  ditierence  of  level  between  the  surface 
of  the  water  in  the  reservoir  and  the  discharge-pipe.  Hence  the 
quantity  of  work  is  estimated  by  the  same  rule,  Equation  (738).  If 
we  omit  for  a  moment  the  consideration  of  friction,  and  take  but  a 
single  elevation  of  the  piston  after  the  water  has  reached  the  dis- 
charge-pipe, 11  will  equal  one,  (p  will  be  zero,  and  that  equation  re- 
duces to 

Fj)  =  G2,5  Up   y  b  ; 

bu-t  G2,5  X  Bp  is  the  quantity  of  fluid  discharged  at  each  double 
stroke  of  the  piston,  and  b  being  the  elevation  of  the  dischai'ge-pipe 
above  the  water  in  the  reservoir,  we  see  that  the  work  will  be  the 
same  as  though  that  amount  of  fluid  had  actually  been  lifted  through 
this  vertical  height,  which,  indeed,  is  the  useful  eflect  of  the  pump 
for  every   double  stroke. 

§  394. — The  Forcing-Pump 
is  a  further  modification  of 
the  simple  sucking-pump.  The 
barrel  B  and  slee|iing-valve 
F  are  placed  upon  the  top 
of  the  sucking-pipe  M.  The 
piston  F  is  without  per- 
foration and  valve,  and  the 
water,  after  being  forced  into 
the  barrel  by  the  atmospheric 
pressure  without,  as  in  the  suck- 
ing-pump, is  driven  by  the  de- 
pression of  the  piston  through 
a  lateral  pipe  //  into  an  air- 
vessel  iV,  at  the  bottom  of 
which  is  a  second  sleeping-valve 
E\  opening,  like  the  first,  up. 
ward.  Through  the  top  of  the 
air-vessel  a  discharge-pipe  K 
passes,  air-tight,  nearly  to   the 


APPLICATIONS.  477 

bottom.  The  Moter,  when  forciMl  into  the  aii--ve3se]  hy  the  de- 
scent of  the  piston,  rises  above  the  hower  end  of  this  pipe, 
confines  and  compresses  the  air,  whicli,  reacting  by  its  elas- 
ticity, forces  the  water  up  the  pipe,  while  the  valve  E'  is  closed  by 
its  own  weight  and  the  pressure  from  above,  as  so(jn  as  the  i)iston 
reaches  the  lower  limit  of  its  play.  A  few  strokes  of  the  piston  will, 
in  general,  be  sufficient  to  raise  water  in  the  pipe  K  to  any  desired 
height,  the  only  limit  being  that  determined  by  the  power  at  com- 
mand and  the  strength  of  the  pump. 

§  395. — -During  the  ascent  of  the  piston,  the  valve  E'  is  closed 
and  E  is  open  ;  the  pressure  upon  the  upper  surface  of  the  piston 
is  that  exerted  by  the  entire  atmosphere ;  the  pressure  upon  the 
lovi'er  surface  is  that  of  the  entire  atmosphere  transmitted  from  the 
surface  of  the  reservoir  through  the  fluid  up  the  pump,  diminished 
by  the  weight  of  the  column  of  water  whose  base  is  the  area  of 
the  piston  and  altitude  the  height  of  the  piston  above  the  surface 
of  the  water  in  the  reservoir  ;  hence,  the  resistance  to  be  overcome 
by  tlKi  power  will  be  the  difference  of  these  pressures,  whieh  is 
obviously  the  weight  of  this  column  of  water.  Denote  the  area 
of  the  piston  by  i>,  its  height  above  the  water  of  the  reservoir  ai 
one  instant  by  y,  and  the  weight  of  a  unit  of  volume  of  the  fluid 
by  w^  then  will  the  resistance  to  be  overcome  at  this  point  of  the 
ascent  be 

w  .B.ij; 

and    the    elementary  cpxantity  of  work   will   be 

w  .  B  .ijdy; 

and  the  whole  work    during    the    ascent  will    be 

^v  .  Bf'\j  dy  r:.w.B.  '^-^  (/  -  y  J  ; 

in  which  y'  and  y^  are  the  distances  of  the  upper  and  lower  limits 
of  the    play  of  the    piston  from    the  water    in    the    reservoir. 

But  B .  [y'  —  y^)  is  the  volume  of  the  barrel  within  the  limits 
of  the  play  of  the  piston,  and  1  (//'  +  ?/,)  is  the  height  of  its  centre 
of   gravitv   above    the    level   of   the   fiuid    in    the    reservoir. 


478  ELEMENTS     OF     ANALYTICAL     MECHANICS. 

y'  -\-  y 
Denoting  the    play  by  -p,  and    making    — - — '-  =  z',  we    have  foi 

the   qi.antity   of  ^york    during   the    ascent, 

Durino-  the  descent  of  the  piston,  the  valve  E  is  closed,  and  E' 
open,  and  as  the  columns  of  the  fluid  m  the  barrel  and  discharge- 
pipe,  below  the  horizontal  plane,  of  the  lower  surface  of  the  pistou, 
will  maintain  each  other  in  equilibrio,  the  resistance  to  be  over- 
come by  the  power  will  be  the  weight  of  a  column  of  fluid  whose 
base  is  the  area  of  the  piston  and  altitude  the  difference  of  level 
between  the  piston  and  point  of  delivery  P ;  and  denoting  by  z. 
the  distance  of  the  central  point  of  the  play  below  the  point  JR. 
we    shall    find,  by  exactly  the    same    process, 

tv  B  p  z,  , 

for  the  quantity  of  work  of  the  motor  during  the  descent  of  the 
piston ;  and  hence  the  quantity  of  work  during  an  entire  double 
stroke  will  be    the   sum  of  these,  or 

But  z'  +  Zi  is  the  height  of  the  point  of  delivery  P  above  the 
surface  of  the  water  in  the  reservoir ;  denoting  this,  as  before,  by 
6,  we   have  * 

%v  Bp  h  ; 

and  calling  the  number  of  double  strokes  ?^,  and  the  whole  quantity 
of  work   §,  we   finally    have 

Q  =  n  w  Bpb. C/SO) 

If  we    make    z^  =  2',    or   h  —  2  z, ,  which  will  give    z,  =  —5    the 

quantity  of  work  during  the  ascent  will  be  equal  to  that  during 
the  descent,  and  thus,  in  the  forcing-pump,  the  work  may  be  equalized 
and  the  motion  made  in  some  degree  regular.  In  the  lifting  and 
sucking-pumps  the  motor  has,  during  the  ascent  of  the  piston,  to 
overcome  the  weight  of  the  entire  column  whose  base  is  equal  to 
the  area  of  the   piston   and   altitude   the   difference  of  level   between 


APPLICATIONS.  47U 

the  water  in  the  reservoir  and  point  of  delivery,  and  being  wholly 
relieved  during  the  descent,  when  the  1  )ad  is  thrown  upon  the 
sleeping-valve  and  its  box,  the  work  becomes  variable,  and  the 
motion    irregular. 

THE    SIPHON. 

§  396. — The  Siphon    is   a  bent  tube  of  unequal    branches,    open   ai 
both  ends,   and    is    used    to  convey    a    liquid 
from   a  higher  to   a   h)\sei'   level,  over  an  in- 
termediate    point     liigher     than     either.      Its  ^... -^^^^^ -f,  ' 

parallel    branches    being  in    a    vertical    plane  pi    I [f'^\     j^ 

and    ])lunged    into  two    liquids    whose    upper  f^''<    J*'    '     '     ;' 

surfaces    are    at    L  M  and    L'  M\    the    fluid  It  '     .;     ,^ 

will    stand    at    the    same    levci     both    witliiu  |;  \    \     m 

and   without   each    branch   of  the    tube    when  jr'-  •' 

a    vent    or     small    opening     is     made    at    0.  IfesS^iia 

If    the    air    be    witlidrawn    from     the    siphon 
thi-ough  this  vent,   the  water  will  rise  in  the 

branches  by  the  atmospheric  pressure  without,  and  when  the  two 
columns  unite  and  the  vent  is  closed,  the  liquid  will  flow  from  the 
resei-voir  A  to  A\  as  long  as  the  level  L'  M'  is  below  L  M,  and  the 
end  of  the  shorter  branch  of  the  siphon  is  below  the  surface  of  the 
liquid    in    the  reservoir    A. 

Tlie  atmospheric  pressures  upon  the  surfaces  L  M  and  L'  M', 
tend  to  force  the  liquid  up  the  two  branches  of  the  tube.  "When 
the  sij  iion  is  filled  with  the  liquid,  each  of  these  pressures  is  coun- 
teracted in  part  by  the  pi-essure  of  the  fluid  column  in  the  branch 
I  if  the  siphon  that  dips  int'.>  the  fluid  upon  which  the  pressure  is 
(xeiled.  The  atmospheric  j)rcssures  are  very  nearly  the  same  for  a 
iifVertHicc  of  level  of  several  feet,  by  reason  of  the  slight  density 
i.f  jiir.  The  pressures  of  the  suspended  columns  of  water  will,  for  the 
sjtme  diflerence  of  level,  differ  considerably,  in  consequence  of  the 
g!-eater  density  of  the  liquid.  The  atmospheric  pressure  opposed 
to  the  weight  of  the  longer  colunui  will  therefore  be  more  counter- 
acted   than    that   opposed    to   the    weight  of    the   shorter,   thus  leaving 


480  ELEMENTS     OF     AXALYTICAL    MECHANICS. 

an  excess  of  pressure  at  the  end  of  the  shorter  branch,  which  will 
produce  the  motion.  Thus,  denote  by  A  the  intensity  of  the  at- 
mospheric pressure  upon  a  surfoce  a  equal  to  that  of  a  cross-section 
of  the  tube;  by  h  the  difference  of  level  between  the  surface  i/ J/ 
and  the  bend  0 ;  by  k'  the  difference  of  level  between  the  same 
point  0  and  the  level  L'  M' ;  by  D  the  density  of  the  liquid; 
and  by  rj  the  force  of  gravity:  then  v»ill  the  pressure,  which  tends 
to    force    the   fluid  up    the    branch    which   dips    below    L  3£,  be 

A  —  a  h  D  [/  ; 

and  that  which  tends  to  force  the  fluid  up  the  branch  immersed 
in    the    other   reservoir,  be 

A  —  ah'  Dg; 
and  subtracting  the  first  from  the  second,   we  find 

al)  (j{h'  -  h), 

for  the  intensity  of  the  force  wliich  urges  the  fluid  within  the 
siphon,  from  the  upper  to   the   lower    reservoir. 

Denote  l)y  I  the  length  of  the  siphon  from  one  level  to  the 
other.  This  will  be  the  distance  over  which  the  above  force  AviJl 
be    instantly    transmitted,    and    the     quantity     of    its    work    will     be 

measured    by  Hi    y  ^   d  ^. 

aDg{h'  -  h)l.  flc4/<L  K  - 

The  mass  moved  will  be  the  fluid  in  the  siphon  which  is  measured 
by  a  ID;  and  if  we  denote  the  velocity  by  V,  we  shall  have,  for  the 
living  force  of  the  moving  mass, 

all).  72; 
whence, 

aDg{h'  -h)l  =  "~^^^; 

and, 

F:=  ^2g{h'  -h); 

from  which  it  appears,  that  the  velocihj  with  u-hich  the  liquid  toill 
Jloiv  through  the  siphon,  is  equal  to  the  square  root  of  twice  the  force 
of  gravity,  into    the   difference   of  level   of   the  fluid  in    the    two   reser- 


APPLICATIONS. 


481 


voirs.  When  the  fluid  in  the  reservoirs  comes  to  the  same  level, 
the   flow    will    cease,  since,  in  that  case,  h'  —  h  —  0. 

§  397. — The  siphon  may  be  employed  to  great  advantao^e  to 
drain  canals,  ponds,  marshes,  and  the  like.  For  this  purpose,  it  may 
be  made  flexible  by  constructing  it 
of  leather,  well  saturated  with 
grease,  lilve  the  common  hose,  and 
furnished  with  internal  hoops  to 
prevent  its  collapsing  by  the  pres- 
sure of  the  external  air.  It  is 
thrown  into  the  water  to  be  drained, 
and  filled ;  when,  the  ends  being 
plugged  up,  it   is  placed  across  the 

ridge  or  bank  over  which  the  water  is  to  be  conveyed ;  the  plugs 
are  then  removed,  the  flow  will  take  j^lace,  and  thus  the  atmos- 
phere will  be  made  literally  to  press  the  water  from  one  basin  to 
another,  over   an   intermediate  ridge. . 

It  is  obvious  that  the  difference  of  level  between  the  bottom  of 
the  basin  to  be  drained  and  the  highest  jDoint  0,  over  which  the 
water  is  to  be  conveyed,  should  never  exceed  the  height  to  which 
water  may  be  supported  in  vacuo  by  the  atmospheric  jjressure  at 
the   place. 


THE    AIR-PUMP. 


§  398. — Air  expands  and  tends  to  diffuse  itself  in  all  directions 
when  the  surrounding  pressure  is  lessened.  By  means  of  this  pro- 
perty, it  may  be  rarefied  and  brought  to  almost  any  degree  of  tenu- 
ity. This  is  accomplished  by  an  instrument  called  the  Air-Pump  or 
Exluiusting  Syringe.  It  will  be  best  understood  by  describing  one 
of  the  simplest  kind.      It  consists,  essentially,  of 

1st.  A  Receiver  H,  or  chamber  from  which  the  exterior  air  is  ex- 
cluded, that  the  air  within  may  be  rarefied.  This  is  commonly  a 
bell-shaped  glass  vessel,  with  ground   edge,  over  which  a   small  quan 

tity  of  grease  is  smeared,  that  no  air  may  pass  through  any  remain- 

31 


482 


ELEMENTS     OF    ANALYTICAL    MECHANICS. 


ing  inequalities  on  its  surface,  and  a  ground  glass  plate  m  n  imbedded 
in  a  metallic  table,  on  which  it  stands. 

2d.  A  Barrel  £, 
or  chamber  into 
which  the  air  in 
the  reservoir  is  to 
expand  itself.  It 
is  a  hollow  cylin- 
der of  metal  or 
glass,  connected 
with  the  receiver 
H  by  the  commu- 
nication offf.     An 

air-tight  piston  P  is  niade  to  move  back   and  forth  in   the  barrel   by 
means  of  the  handle  a. 

3d.  A  Stop-cock  7i,  by  means  of  which  the  communication  between 
the  barrel  and  receiver  is  established  or  cut  oif  at  pleasure.  This 
cock  is  a  conical  piece  of  metal  fitting  air-tight  into  an  aperture 
just  at  the  lower  end  of  the  barrel,  and  is  pierced  in  two  directions; 
one  of  the  perforations  runs  transversely  through,  as  shown  in  the 
first  figure,  and  when  in  this  position  the  communication  between 
the  barrel  and  re- 
ceiver is  estab- 
lished ;  the  second 
perforation  passes 
in  the  direction  of  t^^ — .  !  -/ 

the  axis  from    the  ^J-^~ 

I 
smaller     end,    and 

as     it     approaches 

the   first,  inclines   sideways,  and  runs   out   at   right    angles   to   it,   as 

indicated   in   the    second   figure.      In   this   position   of   the   cock,    the 

communication    between    the   receiver   and   barrel    is    cut    off,    whilst 

that  with  the  external  air  is  opened. 

Now,   suppose    the   piston   at   the   bottom   of   the   barrel,   and   the 

communication    between     the    barrel    and     the    receiver    established ; 

draw  the  piston  back,  the  air    in    tlic    receiver   will    rush    out   in   the 


APPLICATIONS..  4S3 

direction  indicated  by  the  arrow-head,  through  (ne  communication 
o/y,  into  the  vacant  space  within  the  barrel.  The  air  which  now 
occupies  both  the  barrel  and  receiver  is  less  dense  than  when  it  occu- 
pied the  receiver  alone.  Turn  the  cock  a  quarter  round,  the  com- 
munication between  the  receiver  and  barrel  is  cut  off,  and  that  be- 
tween the  latter  and  the  open  air  is  established;  push  the  piston  to 
the  bottom  of  the  barrel  again,  the  air  within  the  barrel  will  be 
delivered  into  the  external  air.  Turn  the  cock  a  quarter  back,  the 
communication  between  the  barrel  and  receiver  is  restored ;  and 
the  same  operation  as  before  being  repeated,  a  certain  quantity  of 
air  will  be  transferred  froni  the  receiver  to  the  exterior  space  nt 
each  double  stroke  of  the  piston. 

To  find  the  degree  of  exhaustion  after  any  number  of  double 
strokes  of  the  piston,  denote  by  JJ  the  density  of  the  air  in  the  re- 
ceiver before  the  operation  begins,  being  the  same  as  that  of  the 
external  air;  by  r  the  capacity  of  the  receiver,  by  6  that  of  the  bar- 
rel, and  by  p  that  of  the  pipe.  At  the  beginning  of  the  operation, 
the  piston  is  at  the  bottom  of  the  i)aiTe],  and  the  internal  air  occu 
pies  the  receiver  and  pipe;  when  the  piston  is  withdrawn  to  the 
opposite  end  of  the  ban-el,  this  same  air  expands  and  occupies  the 
receiver,  pipe,  and  barrel ;  and  as  the  density  of  the  same  bod5''  is 
inversely  proportional  to  the  space  it  occupies,  we  shall   have 

ni  which  x  denotes  the  density  of  the  air  after  tlie  piston  is  di-awn 
back  the  first  time.       From  this  proportion,  we  find 

X  =.  D  • — -. 

The  cock  being  turned  a  quarter  round,  the  piston  pushed  back  to 
the  bottom  of  the  barrel,  and  the  cock  again  turned  to  open  the 
communication  with  the  receiver,  the  operation  is  repeated  iqjon  the 
air  whose  density  is  .r,  and  we  have 

r  +  p  +  i      :     r  ^  p     :  :      D  -        '''  "^  ^'~     :     x' ; 

in  which  x'  is  the  density  after  the  second  backward  motion  -of  the 
piston,  or  after  the  second  double  stroke ;    and   we  find 


4S4 


ELEMENTS  OF  ANALYTICAL  MECHANICS. 


D 


r  +  'p 


.r  +  p  +  y 

and  if  n  denote  the  nuinbor  of  double  strokes  of  the  piston,  and 
Xn  the  c  H-resi^.onding  density  of  tlie  remaining  air,  then  will 

\r  -\-  p  +  h/ 

From  which  it  is  obvious,  that  although  the  density  of  the  air  will 
become  less  and  less  at  every  double  stroke,  yet  it  can  never  be 
reduced  to  nothing,  however  great  n  may  be;  in  other  words,  the 
air  cannot  be  wholly  removed  from  the  receiver  by  the  air-punip. 
The  exhaustion  will  go  on  rapid  1\  in  pioportiou  as  the  barrel  is 
large  as  comi^ared  with  the  receiver  and  pipe,  and  after  a  few  double 
strokes,  the  rarefaction  will  be  sufficient  for  all  prj'.ctical  purposes. 
Suppose,  for  example,  the  receiver  to  contain  19  units  of  volume,  the 
pipe  1,  and   the  barrel   10;  then  will 

r  +  p        _  ';20  _  2  _ 
r  +  j>j  +  6  ■"  ;iU  ~  ^  ■ 

and    suppose   4    double    strokes    of  the  piston  ;    then  will  n  -—  4,  and 
(-4-^y  =  (§)■'==-==  0,107,   nearly  ; 

that  is,  after  4  double  strokes,  the  density  of  the  remaining  air  will 
be  but  about  two  tenths  of  the  original  density.  With  the  best 
machines,  the    aii-    may    be   rarefied  fi-om    four   to    six  hundred   times. 

The  degree  of  rarefaction  is  indicated  in  a  very 
simple  manner  by  what  are  called  (jauges.  These 
not  only  indicate  the  condition  of  the  air  in  the 
receiver,  Init    also    warn    tlie    operator  of  any  leakage  Vf 

that  may  take  place  either  at  the  edge  of  the  receiver 
or  in  the  joints  of  the  instrument.  The  mode  in 
which   the  gauge  ;icts,  will  be  readily  understood  fioni  'it 

the  dirciisi-ion  of  the  barometer;  it  will  be  sulli- 
cient  here    simply  to  indicate  its  construction.      In  its  '^^  ^ 

nn)re  perfect  Ibrm,  it  consists  of  a  glass  tube,  about  60  inches  long, 
bent  in  the  middlt;  till  the  straight  portions  are  parallel  to  each 
other ;  one  end  is  closed,  and    the   branch   terminating   in   this    end  is 


APPLICATIONS. 


485 


fille'd  with  mercury.  A  scale  of  equal  parts  is  placed  between  the 
liraiiches.  having  its  zero  at  a  point  midway  from  the  top  to  the 
bottom,  the  numbers  of  the  scale  increasing  in  both  directions.  It 
is  placed  so  that  the  branches  of  the  tube  shall  be  vertical,  Avith 
its  ends  upward,  and  inclosed  in  an  inverted  glass  vessel,  which 
communicates  with  the  receiver  of  the  air-pump. 

Repeated  attempts  have  been  made  to  bring  the  air  pump  to 
still  higher  degrees  of  perfection  since  its  first  invention.  Self-acting 
valves,  opening  and  shutting  by  the  elastic  force  of  the  air.  have 
been  used  instead  of  cocks.  Two  barrels  have  iieen  employed  in- 
stead of  one,  so  that  an  uninterrupted  and  moie  rapid  rarefaction 
of  the  air  is  brought  about,  the  piston  in  one  Ijari'cl  being  made 
to  ascend  while  that  of  the    other  descends.     The   most  serious  defect 


was  that  by  which  a  portion  of  the  air  was  retained  between  the 
piston  and  the  bottom  of  the  barrel.  This  intervening  space  is  filled 
with   air    of    the   ordinary    density    at     each   descent   of    the   piston ; 


4S6  ELEMENTS     OF     ANALYTICAL     MECHANICS. 

when  the  cock  is  turned,  and  the  communication  re-established  with 
the  receiver,  this  air  forces  its  way  in  and  diminishes  the  rarefac- 
tion already  attained.  If  the  air  in  the  receiver  is  so  lar  rarelied, 
thtit  one  stroke  of  the  piston  will  only  raise  such  a  quantity  as 
equals  the  air  contained  in  this  space,  it  is  plain  that  no  further 
exhaustion  can  Le  etfected  by  continuing  to  pump.  This  limit  to 
rarefaction  will  be  arrived  at  the  sooner,  in  proportic^n  as  the 
space  below  the  piston  is  larger ;  and  one  chief  point  in  the  im- 
provements has  been  to  diminish  this  space  as  much  as  possible. 
A  B  is  a  highly  polished  cylinder  of  glass,  which  serves  as  the  bar- 
rel of  the  pump  ;  within  it  the  piston  works  perfectly  air-tight.  The 
piston  consists  of  washers  of  leather  soaked  in  oil,  or  of  cork 
covered  with  a  leather  cap,  and  tied  together  about  the  lower  end 
C  of  the  piston-rod  by  means  of  two  parallel  metal  plates.  The 
piston-rod  Cb,  which  is  toothed,  is  elevated  and  depressed  by  means 
of  a  cog-wheel  turned  by  the  liandle  M.  If  a  thin  iilm  of  oil  be 
poured  upon  the  upper  surface  of  the  piston  the  friction  will  be 
lessened,  and  the  whole  will  be  rendered  more  air-tight.  To  diminish 
to  the  utmost  the  space  between  the  bottom  of  the  barrel  and  the 
[lislon-rod,  the  form  of  a  truncated  cone  is  given  to  the  latter,  so 
that  its  extremity  may  be  brought  as  nearly  as  possi'bUi  into  abso- 
lute contact  with  the  cock  E ;  this  space  is  therefore  rendered  indeli- 
nitely  snuill,  the  oozing  of  the  oil  down  the  barrel  contributing  ^till 
further  to  lessen  it.  The  exchange  cock  E  has  the  double  bore 
already  described,  and  is  turned  by  a  short  lever,  to  which  motion 
is  c<jmmunicated  by  a  rod  c  d.  The  communication  G II  is  carried 
to  the  two  plates  /  and  K.  on  one  or  both  of  which  receivers  may 
be  placed  ;  the  two  cocks  iV  and  0  below  these  ])lates,  serve  to  cut 
oil"  the  I'arefied  air  within  the  receivers  when  it  is  desii-ed  to  leave 
them  for  any  k'Ugth  of  time.  The  cock  0  is  also  an  exchange-cock, 
so   as   to   admit   the   e.vternal   air   into   the    receiver-. 

Pumps  thus  constructed  have  advantages  over  such  as  work 
with  valves,  in  tliat  llu'V  last  longer,  exhaust  better,  and  may  be 
employed  as  cnudiMisers  when  suitable  receivers  are  provided,  by 
merely  reversing  the  operations  of  the  exchange  valve  during  the 
motion  of  the  piston. 


TABLES. 


TABLE    1. 

THE   TENACITIES   OF    DIFEEEEKT   SUBSTANCES,  AND   THE  KESISTANCES 
WHICH  THEY  OPPOSE  TO  DIEEOT  COMPEESSION.— See  §  269. 


SUBSTANCES    EXPERIMENTED    ON. 


Wrought-iron,  in  wire  from  l-20th 
to  l-30tli  of  an  inch  in  diame- 
ter   

in  wire,  1-lOth  of  an  inch  •      •      • 
in  bars,  Kussian  (mean)   • 
English  (mean)    • 

hammered 

rolled  in  sheets,  and  cut  length- 
wise       

ditto,  cut  crosswise    •     •     • 

in  chains,  oval  links  6  in.  clear, 

iron  H  in.  diameter  .    ■      .     • 

ditto,  Brunton's,  with  stay  across 

link.    ...'..... 

Cast  Iron,  quality  No.  1      .     •     •     • 


Steel,  cast 

east  and  tilted  ... 

blistered  and  hammered 

shear  

raw 

Damascus     .... 
ditto,  once  refined    • 
ditto,  twice  refined  • 
Copper,  cast 

hammered    .     .     .     . 

sheet 

wire 

Platinum  wire 

Silver,  cast 

wire 

Gold,  cast 

wire 

Brass,  yellow  (fine)  ... 
Gun  metal  (hard)  •  •  • 
Tin,  cast    

wire 

Lead,  cast 

milled  sheet 

wil'C 


6o  to  91 

36  to  43 

27 

25.V 

3o 

14 

18 

2Sh 
25 

6  to  7J 
6  to  8 
6  to  9} 

44 

60 

59^ 

57 
5o 
3i 
36 
44 


27!- 

n 

iS 
17 

9 
14 

8 
16 

2 

3 
4-5thj 

III 


Lame 

Telford 
Lame 

Brunei 
Mitis 

Bi-o-./n 

Bai-low 
Hodekinson 


;:\iitis 

licnnie 


Mitis 

Eennic 
Kingston 
C.uvton 


Trcdgold 
G nylon 


J3  c  rj2 


38  to  41 
37  to  48 
5 1    to  65 


52 
46 


llodifkinson 


Eeunie 


•The  striingest  quality  01  cast  iron,  is  a  Scotch  iron  known  as  the  Devon  Hot  Blast,  No.  3:  Us  tenaci- 
ty is  9J  tons  per  square  inch,  and  its  resistance  to  conjprcssion  65  tons.  The  experiments  of  Major 
Wude  on  the  gun  iron  at  West  Point  Foundry,  and  at  IJoslon,  give  results  as  high  as  10  to  IC  tons,  and 
on  small  cast  bars,  as  high  as  17  tons.— See  Ordnance  Manual,  1850,  p.  402- 


TABLE    I. 


489 


TABLE  I — continued. 


SUBSTANCES     EXPERIMENTED    ON. 


Stone,  slate  (Welsli)      •     • 

Marble  (white)  • 

Giviy 

Portland 

Craigleith  freestone     • 

Brauiley  Fall  sandstone 

Cornish  granite 

Peterhead  ditto 

Linicstcuie  (compact  blk) 

Purbeck 

Aberdeen  granite    • 
Brick,  pale  red    .... 

red 

Hammersmith  (pavior's) 
ditto      (burnt)  • 
Chalk  ........ 

Plaster  of  Paris  .... 

Glass,  plate 

Bone  (ox) 

Hemp  fibres  glued  together 
Slrips  of  paper  glued  togethe 
vVood,  Box,  spec,  gravity  • 

Ash     .... 

Teak-        .      .      . 

Beech 

Oak     .... 

Ditto   .... 

Fir       .... 

Pear    .... 

Mahogany     •   '  ■ 

Elm     .... 

Pine,  American 

Deal,  white   • 


,6 

,9 

,7 

.92 

,77 

,6 

J  646 

,637 


5,7 
4 


,o3 


41 
i3 

9 


5 

4t 

3i 

6 

6 

6 


Barlow 


1.4 
1,6 

2-4 
2,7 

2:8 

3,7 
4 


,56 
,8 


1,7 


,57 

,73 
,86 


Keniiie 


490 


TABLE    II. 


TABLE  n. 

OF  THE  DENSITIES  AND  VOLUMES  OF  WATER  AT  DIFFERENT  DEGREES 
OF  HEAT,  (ACCORDING  TO  STAMPFER),  FOR  EVERY  2i  DEGREES  OF 
FAHRENIIEITS   SCALE.— See  §  276. 

{.Jahrbut     des  Pulytechnischen  Institutes  in   IVein,  Bd.  16,  .S.  70). 


t 

Teniperalure. 

Density. 

Diir. 

V 
Viihiine. 

Diir. 

0 

32,00 

0,999887 

1,0001 13 

34.25 

0,999950 

63 

i,oooo5o 

63 

36,5o 

0,999988 

38 

1,000012 

38 

38,75 

1,000000 

12 

1,000000 

12 

4i,oo 

0,999988 

12 

1,000012 

12 

43,23 

0,999932 

35 

1,000047 

35 

45,5o 

0,999894 

58 

1,000106 

59 

47,75 

0,999813 

81 

1,000187 

81 

5o.oo 

0,999711 

102 

1,000289 

102 

02,25 

0,999387 

124 

I. 000413 

124 

54>5o 

0,999442 

145 

i,ooo558 

145 

56,75 

0,999278 

164 

1,000723 

i65 

59,00 

0, 99909') 

iS3 

1,000906 

1 83 

61, 25 

0,998893 

202 

1,001108 

202 

63. 5o 

0,998673 

220 

1,00(329 

221 

65,75 

0,998435 

238 

1,001367 

238 

68,00 

0,998180 

235 

1,001822 

255 

70,25 

0,997909 

271 

1,002095 

273 

72,50 

0.997622 

287 

1J002384 

289 

74,75 

0,997320 

3o2 

1,002687 

3oi 

77,00 

0,997003 

3.7 

i.oo3oo5 

3iS 

79,23 

0,996673 

33o 

i,oo3338 

333 

8i,5o 

0,996329 

344 

1, 003685 

347 

83,75 

0,99597 1 

358 

1,004045 

36o 

86,00 

0,993601 

370 

1,004418 

373 

88,25 

0,995219 

382 

1,004804 

386 

90,50 

0,994823 

394 

1.003202 

398 

92,75 

0.994420 

4o5 

i,oo56i2 

410 

95,00 

0,994004 

416 

i,oo6o32 

420 

97,25 

0.99J379 

425 

1,006462 

43o 

99, 5o 

0,993145 

434 

1,006902 

440 

With  this  trilile  it  is  easy  to  fiiiil  the  spciiilic  pr.ivity  liy  means  of  water  at  any  temperature. 
Suppose,  for  e.xample,  the  specific  gravity  S'  in  Equntion  (4.J6),  had  been  found  at  the  tempera- 
ture of  590,  then  woiihl  D„  in  that  equation  lie  0,90909,'),  and  the  specific  gravity  of  the  body 
referred   to  water  at  its    greatest  density,  would   be  given   by 

■S  =  S'  X  0,999095. 


TABLE    III. 


491 


TABLE  III. 

F  THE  SPECiriC  GRAVITIES  OF  SOME  OF  THE  MOST  IMPOKTANT  BODIES. 
[The  density  of  distilled   wuter   is   reclioiied  in   this   Table   sit   its   niaxiinum   38J0  I'\  =  1,000]. 


Name  tif  tlie  Body. 


Specific  Gravity. 


I.    SOLID  BODIES. 

(1)  Mktals. 

Antimony  (of  the  laboi'iitofy)  • 
Brass       ..... 
Bronze  for  cannon,  according  to  Lieut.  Mttzka 
Ditto,  mean     .... 
Copper,  melted 
Ditto,  liiuiyiiered 
])itiii,  wire-drawn    • 
Gold,  melted    .... 
Ditto,  liammered 
Iron,  wrouglit 
Ditto,  cast,  a  mean  • 
Ditto,  gray       .... 
Ditto,  white    .... 
Ditto  for  cannon,  a  mean 
Leai.l,  |iure  melted  • 
Ditto,  tiattened 
I'laiiiuiin,  ntitive 
Ditto,  melted  .         .         •         • 
IJitto,  liammered  and  wire-dr;iwn 
tiiticksilver,  at  32°  Fahr. 
Silver,  jnire  melted 
Oittii,  hatnmered 
Steel,  cast         .... 
■  Ditto,  wrought 
Ditto,  mucli  hardened 
Ditto,  slik^htly 
Ti:i,  chemically  pure 
Ditto,  hammered 
Ditto,  Bohemian  and  Saxon 
Ditto,  English 

Zinc,  melted    .... 
Ditto,  rolled    .... 

(2)  Building  Stones, 

Alabaster  .... 

Basalt  ..... 
Dole  rite  ..... 
Gneiss  ..... 
Granite  .  .  .  .  • 
Hornblende  .... 
Limestone,  various  kinds 
Bhonolite  .... 

Porphyry         .... 

Quartz 

Sandstone,  various  kinds,  a  mean 
Stones  for  building  • 
Syenite    ..... 
Traehvte  .... 

Brick'      ..... 


4,2          — 

4,7 

7,6        - 

8,8 

8,414    — 

8,974 

8,738 

7,788    - 

8,726 

8,878    — 

8,9 

8,78 

19.238    — 

19,253 

19,361     — 

19,6 

7,207     — 

7.788 

7,231 

7,2 

7,5 

7,21         — 

7-30 

ii.33o3 

ii,3SS 

16,0        — 

18,94 

20,855 

21,25 

i3,568    — 

13,598 

10,474 

10, 5i       — 

10,622 

7,9'9 

7,840 

7,Si8 

7,833 

7,291 

7,299    — 

7,475 

7,3iJ 

7,291 

6,861     — 

1.2i5 

7,191 

2,7         — 

3,0 

2,8 

3,1 

2,72         — 

2,93 

2,5           — 

2,9 

2,5           — 

2,66 

2,9        — 

3,1 

2,64     — 

2,72 

5,5l          

2,69 

2,4        — 

2,6 

2,56      — 

2.75 

2,2           — 

2,5 

1,66      — 

2,62 

2,5           — 

3, 

1,41 


492 


TABLE    III. 


TABLE   lU—Contmued 


Name  of  the  Body. 


Specific  Gravity. 


I.    SOLID  BODIES. 


(3)  Woods. 


Alder 

Asli 

Aspeii 

Eircli 

Box 

Elm 

Eir 

Hornbeam 

Ilorjie-cliestuiit 

Larch       . 

Lime 

Miiplc 

Oak 

Ditto,  aiiotlicr  specimen 

Pine,  Plnus  Abies  Picea 

Ditto,  Pinvs  Sylvestris 

Poplar  (Italian) 

Willow    • 

Ditto,  white     • 


(4)  Vaeious  Solid  Bodii; 


Charcoal,  of  cork 

Ditto,  soft  wood 

Ditto,  oak 

Coal 

Coke 

Earth,  common 

rong'hsand 

rouffh  earth,  with  grr 

moist  sand 

gravelly  soil 

clay  . 

clay  or  loam,  with  gravel 
Flint,  dark 
Ditto,  wliite     • 
Gunpowder,  loosely  filled 

coarse  powder  • 

musket  ditto     • 
Ditto,  slitrhtly  shaken  do 

musket-powder 
Ditto,  solid      . 
lee  • 

Lime,  unslacked 
Kcsin,  common 
Kock-salt 
Saltpetre,  melted 
Ditto,  crystallized 
Slate-jiencil 
Sulphur  • 
Tallow     • 
Turpentine 
^^'ax,  white 
Ditto,  yellow   • 
Ditto,  shoemaker' 


^h-ffl!ril. 

Prv. 

0,8371 

o,5ooi 

0.90.36 

0,6440 

0,7654 

o.43o2 

0,9012 

0,6274 

0,9822 

0,3907 

0.9476 

0.3474 

0,8941 

0,3330 

0,9402 

0,7695 

0,8614 

0,3749 

0.9206 

0,4735 

0,8170 

0,4390 

o,9o36 

0.6092 

1,0494 

0,6777 

1.0754 

0.7073 

0,8699 

0,4716 

0,9121 

o,55o2 

0,7634 

0.3931 

0,7133 

0,5289 

0,9859 

0.4873 

0,1 

0,28 
1,573 

1,232 

1,865 
1,48 
1,92 
2,02 

2,03 
2,07 
2.l5 

2,48 

2,542 

2,741 

0,886 
0,992 

1,069 
2,248 
0,916 
1.842 

I.oSg 
2,237 
2,745 
1,900 

1,8 

1,92 

0,942 

0,991 

0,969 

o,o65 

o;897 


0,44 
i,5io 


2,563 
0.9268 


2,24 
1,99 


T  A  B  L  E     Til. 


493 


TABLE   lU—Contmuecl 


X;uiie  <ir  tlie  Body. 

specific 

iravily. 

II.  LIQUIDS. 

Acid,  acetic 

I,o63 

Dilto,  iniiriatlc 

1,21  1 

I)iit(),  iiiiric,  coiicoiti'ated 

1.52  1     - 

-      1,522 

Ditto,  suliiiiur'.o,  lCiiL'li>li 

1. 845 

Ditto,  coiiceutratuil  (Xonlli.)    • 

. 

i,b6o 

Alciihdl,  IVue  from  water 

0,792 

Ditto,  (.Miiimi'ii 

0.824     - 

-     0,79 

Aiimiouiac,  ru]ui(l     ■ 

0.875 

Aijualortis,  double  • 

1,3  JO 

Dilto,  single     • 

1,200 

Beer 

1.023     - 

-     i,o3i 

El  her,  acetic    • 

0-866 

Ditto,  iuiiriatie 

0,845     - 

-     o,"74 

Dilto,  nitric     • 

0.886 

Ditto,  sulpliuiio 

0.715 

Oil,  linseed      • 

0,928     - 

-     0,953 

])ilto,  (ilive      • 

0.915 

Ditto,  turiientine     • 

0.792     - 

-     0.R91 

Ditto,  whale    • 

0.923 

Quicksilver      • 

1 3. 568     - 

-  13,598 

Water,  .listilled 

1 ,000 

Ditto,  rain 

1. 001 3 

Ditto,  sea 

1,0265   - 

-     1.028 

Wine 

0.992   - 

-     1  o]8 

III.  GASES. 

\V.tfr=  1. 

n^To.u- 

'lY'iiip.  Sf^o  p. 

r.  in.=:i-10 

.^Vtmospliericair  =  yig  —    • 

o,ooi3o 

1 .0000 

Carbonic  acid  gas 

0  00 ! 98 

:  .^24c 

( 'ai-i)onic  oxide  gas   • 

0  001 26 

0  9J6Q 

Carbureted  hydrogen,  a  maximum 

0,00127 

oo7«4 

Ditto,  from  Coals      • 

\ 

0.00039 

0  3  000 

1 

0.00085 

0  5596 

Chlorine  • 

0,0032I 

2  4700 

Ilydriolicgas- 

0.00377 

4  44  Jo 

IIydi-oL;-en 

0.0000895 

o'.o6S.s 

Hvdrosulpluiric  acid  gas 

o.ooi55 

1  1912 

M'lirlatic  acid  gas      •  ' 

0  00162 

1-24-4 

Nitrogen 

0,00127 

0  9-60 

Oxygen   • 

0.00143 

I  1026 

riiosphureted  livdrogen  , 

]:as     • 

0.001 13 

o.''!70o 

Steam  at  212°  Fahr.  ' 

0.000S2 

0  6235 

Sulrihunuis  acid  sas 

0,00292 

2.2470 

L 

49-1 


TABLE  IV. 


TABLE  lY. 

TABLE  FOE  FINDING  ALTITUDES.— See  §  284. 


Delncheil  Tlicrmoineter. 

t,+t' 

A 

t^  +  t' 

A 

t,+t' 

A 

tj  +  i' 

A 

40 

4,7689067 

75 

4,7859208 

no 

4,8022936 

145 

4,8180714 

41 

,7694021 

76 

,7863973 

III 

,8027323 

146 

,8i85i4o 

42 

,7698971 

11 

,7868733 

112 

,8032109 

147 

,8189559 
,8193975 

43 

,7703911 

78 

,7873487 

ii3 

,8036687 

148 

44 

,7708831 

79 

,7878236 

114 

,8041561 

149 

,8198387 

45 

,7713785 

80 

,7882979 

ii5 

,8o45S3o 

i5o 

,8202794 

46 

,7718711 

81 

,7887719 

116 

,8o5o395 

i5i 

,8207196 

47 

,7723633 

82 

,7892451 

117 

,8054953 

l52 

,8211594  ! 

48 

,7728548 

83 

,7897180 

118 

,8059309 

1 53 

,8213998 

49 

,7733457 

84 

,7901903 

119 

,8064038 

1 54 

,8220377 

5o 

,7738363 

85 

,7906621 

120 

.8068604 

i55 

,8224761 

5i 

,7743261 

86 

,7911335 

121 

,8073144 

136 

,8229141 

52 

,7748153 

87 

,79'6o42 

122 

,8077680 

1 57 

.8233517 

53 

,7753042 

88 

,7920745 

123 

,8082211 

i58 

,8237888 

54 

,7757925 

89 

,7925441 

124 

,8086737 

139 

,8242256 

55 

,7762902 

90 

,793oi35 

I  125 

,8091238 

160 

,8246618 

56 

,7767674 

9' 

,7934822 

126 

,8095776 

161 

,8230976 

57 

,7772540 

92 

,7939504 

127 

,8100287 

162 

,8255331 

58 

,7777400 

93 

,7944182 

128 

,8104795 

i63 

,8239680 

59 

,7782206 

94 

,7948854 

129 

,8109298 

164 

,8264024 

6o 

,7787105 

95 

,7953521 

i3o 

,8113796 

165 

,8268365 

6i 

,779'949 

96 

,7938184 

i3i 

,8118290 

166 

,8272701 

62 

,7796788 

97 

,7962841 

132 

,8122778 

167 
]68 

,8277034 

63 

.7S01622 

98 

,7967493 

i33 

,8127263 

.8281362 

64 

,7806430 

99 

,7972141 

i34 

,8i3i742 

169 

,8285685 

65 

,7811272 

100 

,7976784 

i35 

,8i362i6 

170 

.8290003 

66 

,7816090 

lOI 

,7981421 

i36 

,8140688 

171 

,8294319 

67 

,7^20902 

102 

,7986054 

i37 

,8i45i53 

172 

,8298629 

68 

,7825709 

io3 

,7990681 

i38 

,8149614 

173 

,8302937 

69 

,783o5n 

104 

,79953o3 

139 

,8154070 

174 

,8307238 

70 

,7835306 

io5 

,7999921 
,8004333 

140 

,81 58523 

175 

,83 1 1 536 

71 

,7840098 

106 

141 

,8162970 

176 

,83i583o 

72 

,7844883 

107 

,8009142 

142 

,8167413 

177 

,8320119 

73 

,7849664 

108 

,8018744 

143 

,8171852 

178 

,8324404 

74 

4,7854438 

109 

4,8018343 

144 

4,8176285 

179 

4,8328686 

TABLE   lY. 


495 


/ 1 


TABLE  lY — continued. 

■\V1TII  THE   BAEUMETEE— See  S  284. 


1  ^ 


LiUituile. 

Attached  'riicriiii 

meter. 

^ 

B 

T~T' 

C 

0 

0° 

0,0011689 

^, 

_i_ 

3 

,0011624 

0'3 

0,0000000 

0.0000000 

6 

,0011433 

I 

,0000434 

9,9999566 

9 

,0011 117 

2 

,0000869 

,999913 1 

la 

,0010679 

3 

,000 1 Jo^ 

,9998697 

ID 

,0010124 

4 

,0001738 

,9998263 

i8 

,0009439 

5 

,0002172 

,9097829 

21 

,0008689 

6 

,0002607 

,9997395 

24 

,0007825 

7 

,ooo3o4i 

,9996961 

27 

,0006874 

8 

,0003476 

,9996327 

3o 

,ooo5S4>J 

9 

,0003910 

,9996093 

33 

,000475a 

10 

,0004345 

,9993639 

36 

,000,36 1 5 

II 

,0004780 

,9995223 

39 

,0002433 

12 

,000521 5 

,9994792 

42 

,0001223 

i3 

,ooo565o 

,9994358 

43 

,0000000 

14 

,0006084 

,9993924 

48 

9,9998770 

i5 

,0006519 

,9993490 

49 

,9998372 

16 

,0006954 

,9993057 

5o 

,999707 

17 

,0007389 

,9992623 

5i 

,9997366 

18 

,0007824 

,9992190 

52 

,9997167 

'9 

,0008259 

,9991736 

53 

,9996772 

20 

,0008693 

,9991323 

54 

,  99963  :ii 

21 

,0009130 

,99908^9 

55 

,999)90:') 

22 

,ooog565 

,9990436 

56 

,999)6 1  i 

23 

,0010000 

,9990023 

57 

,9995237 

24 

,0010436  ■ 

,9989389 

58 

,9994h6(5 

25 

,0010871 

,9989136 

59 

,9994302 

26 

,ooii3o6 

,9988723 

60 

,9994144 

27 

,0011742 

,9988290 

63 

,9993 1 1 5 

28 

,0012177 

,9987837 

66 

,9992:61 

29 

,0012613 

,9987424 

69 

,9991293 

3o 

,ooi3o48 

,9986991 

i5 

.99S9S52 

3i 

0,0013484 

9,9986558 

81 

,99S,s854 

90 

9,99j83oo 

49H 


TABLE    V. 


TABLE   \". 

COEFFICIENT  VALUES,  FOE  THE  DISCHARGE  OF  FLUIDS  THKOUGH  THIN 
PLATES,  THE  OEIFICES  BEING  REilOTE  FROM  THE  LATERAL  FACES 
OF  THE  VESSEL— See  §  300. 


Values 

of  the  coelticients  f.ir  nriti 

ees  wliose  sn 

iiUe^i  iliinsn 

»ions  i.T    ^ 

Head  of  fluid 

<liaiiicte 

rs  are — 

above  the 
centre  of  the 

orilice,  in  feet. 

ft. 

/' 

/«• 

ft. 

ft- 

ft 

0,66 

0,33 

0,16 

0,08 

0,07 

o,o3 

o.o3 

0,700 

0,07 

0,627 

0,660 

0,696 

0,1 3 

0.618 

0,632 

0,657 

0.685 

0,20 

0,5g2 

0,620 

0,640 

0,656 

0.677 

0,26 

0.602 

0.625 

0.638 

0,655 

0,672 

0,33 

0,593 

0,608 

o,6Jo 

0.637 

0.655 

0,667 

0,66 

0,596 

o,6i3 

0,63 1 

0634 

0.634 

0,655 

1,00 

0,601 

0,617 

O,()j0 

0.632 

0.644 

o,65o 

1,64 

0,602 

0,617 

0,628 

o,63o 

0.640 

0.644 

-3,28 

o,6o5 

o,6i5 

0.626 

0,628 

0,633 

0,632 

5,00 

o,6o3 

0,612 

0,620 

0,620 

0,621 

0,618 

6,65 

0,602 

0,610 

0,61 5 

o.6i5 

0,6 10' 

0,610 

32,75 

0,600 

0,600 

0,600 

0,600 

0,600 

0,600 

In  the  iiistiincc  of  ;:>is,  the  ■.'Ciieriiting  head  is  always  greater  than  G,G5  ft.,  and  llie  cneiiicii'iit  (1,6, 
or  0,61,  is  taken  ia  ail  cases. 

For  orifices  lai'iier  tliaii  0,06  ft,,  llie  cnefiicients  .are  taken  as  for  this  dimension  ;  for  orifices  smaller 
than  0,03  ft,,  the  coellicients  are  the  same  as  for  this  latter ;  finally,  for  orifices  between  those  of  the 
table,  we  take  coeflicients  whose  values  are  a  mean  between  the  latter,  corresponding  to  the  given  head. 


TABLE   VI. 


497 


TABLE  YI. 

EXPER3IENTS  ON  FKIOTION,  WITHOUT  UNGUENTS.    BY  M.  MOEIN. 

The  surfaces  of  friction  were  varied  from  o,o3336  to  2,7987  square  feet,  tlie  pressures  from 
88  lbs.  to  22a5  lbs.,  and  the  velocities  from  a  sc.ircely  perceptible  motion  tu  9,84  feet  per 
second.  The  surflices  of  wood  were  planed,  and  those  of  metal  filed  and  polished  with  tlie 
greatest  care,  and  carefully  wiped  afier  e\L;-y  experiment.  The  presence  of  unguents  was 
especially  guarded  against..— See  §  355. 


Friction  of 

I'riction  ( 

F 

SURFACES  OF  COiNTACT. 

Motion.* 

UUIKS 

CliNCL 

•t 

n  c 

6 

ti 

OJ 

•r  Z 

c 

c  r 

i.,  .^ 

'i 

- 1 

|| 

•= 

*i-S 

~s, 

■= 

5. -7 

C  t_ 

0  = 

5<^ 

Co 

^ 

~  6 

Oak  upon  oak,  the  direction  of  the  fibres  ( 
being  parallel  to  the  motion      •     •     •  f 

0,478 

2D' 

33' 

0,625 

32^^ 

I' 

Oak  upon  oak,  the  directions  of  the  fibres  " 

of  the  moving  surface  being  perpen- 
dicular to  those  of  the  quiescent  sur-  [ 

0,824 

•7 

58 

0,540 

28 

23 

face  and  to  the  direction  of  the  motionf  J 
Oak  upon  oak,  the  fibres  of  tliebotli  sur-  1 

faces  being  perpendicular  to  the  dircc-  I 

0,336 

18 

35 

tion  of  the  motion \ 

Oak  upon  oak,  tlie  fibres  of  the  moving "] 

surface  being  perpendicular  tn  tliesui-^ 
lace  of  contact,  and  those  of  the  surface  I 

0,192 

10 

52 

0,271 

ID 

10 

at  rest  parallel  to  the  direction  of  the 

motion J 

Oak  upon  oak,  the  fibres  of  both  surfaces  1 

being  perpendicular  to  the  surface  of  I 

0,43 

23 

17 

contact,  or  the  pieces  end  to  end   •      •  ) 

Elm  upon  oak,  the  direction  of  the  fibres  | 
being  parallel  to  the  motion      •      •      •  f 

0,432 

23 

2; 

0,694 

34 

46 

Oak  upon  elm,  ditto§ 

0,246 

i3 

5c 

0,376 

20 

37 

Elm  upon  oak,  the  fibres  of  the  moving  ~l 

surface  (the  elm)  being  perpendicular  to  1 
those  of  the  quiescent  surface  (tlie  oak)  f 

o,45o 

24 

16 

0,570 

29 

41 

and  to  the  dii-ection  of  the  motion-      •  J 

Ash  upon  oak,  the  fibres  of  both  surfaces  i 

being  parallel  to  the  direction   of  the  > 

0,400 

21 

49 

0,570 

29 

41 

motion ) 

Fir  upon  oak,  the  fibres  of  both  surfaces  j 

being  parallel  to  the  direction  of  the  |- 

0.355 

19 

33 

0,020 

27 

29 

ir.otiou ) 

Beach  upon  oak,  ditto 

o,36o 

'9 

48 

0,53 

27 

56 

Wild  pear-tree  upon  oak,  ditto     • 

0,370 

20 

'9 

0,440 

2j 

45 

Service-tree  upon  oak,  ditto    .... 

0,400 

21 

49 

0,570 

'i'i 

41 

Wrought  iron  upon  oak,  dittof   • 

0,619 

3i 

47 

0,619 

3i 

47 

*  Tlic  friction  in  this  case  varie.s  but  very  sligtitly  fioni  the  iiienn. 

t  Tlie  IViftiDn  in  this  Ciise  varies  consideraljly  from  the  mean.  In  all  the  cvjicriments  the  surface  s 
had  beon  15  minutes  in  contact. 

t  The  (lim(■n^il)ns  nf  ihe  surfaces  of  contact  were  in  this  experiment  ,947  pqntire  feet,  ami  the  results 
were  ntioly  uniliirm.  When  tlie  dimensions  were  diminished  to  ,043,  a  tearing  (if  tlie  filire  bccnnie  appa- 
rent in  tlic'case  (if  iiiiition,  and  there  were  symptoms  of  the  comliuslioii  of  the  wood  :  from  these  cir- 
cunistaiices  there  resulted  an  irregularity  in  the  friction  indicatiyi  of  excessive  pressure. 

(S  It  is  worthy  of  remark  that  the  friction  of  oaJ<  upon  elm  islnii  five-ninths  of  that  of  elm  upon  oak. 

|"|  In  the  experiments  hi  which  one  of  the  surfaces  was  of  metal,  small  parliclesof  the  metal  becaii, 
after  a  time,  to  be  apparent  upon  the  wood,  giving;  it  a  polished  metallic  .appearance  ;  tliese  were  atevery 
experiiuen  wiped  off;  they  indicated  a  wearini"  of  the  metal.  The  friction  of  motion  I'.nd  that  of  quies- 
cence, in  these  experiments,  coincided.    The  results  were  remarkably  unif(  1111. 

32 


498 


TABLE    VI. 


TABLE  YI — continued. 


SURFACES  OF  CO.NTACT. 


Friction  of 
Motion. 


Wrcught  irou  upon   oak,  the   surfaces  ) 
being  greased  and  well  wetted-     •      •  j 

Wrought  iron  upon  elm 

Wrought  iron  upon  cast  iron,  the  fibres  | 

of  the  iron  being  parallel  to  the  motion  ( 
Wrought  iron  upon   wrought  iron,  tlie  ) 

fibres  of  both   surfaces  being  parallel  ,- 

to  the  motion ) 

Cast  iron  upon  oak,  ditto 

Ditto,  tlie  surfaces    being  greased   and  / 

wetted f 

Cast  iron  upon  elm 

Cast  iron  upon  cast  iron 

Ditto,  water  being  interposed   between  ) 

the  surfaces j 

Cast  iron  upon  brass 

Oak  upon  cast  iron,  the  fibres  of  the  wood  i 

being  perpendicular  to  the   direction  > 

of  the  motion ) 

Hornbeam  upon  cast  iron — fibres  paral-  / 

lei  to  motion j 

Wild   pear-tree   upon   cast  iron — fibre?  I 

parallel  to  the  motion \ 

Steel  upon  cast  iron 

Steel  upon  brass 

Yellou"  copi^er  upon  cast  iron  .... 
Ditto  oak      .... 

Brass  upon  cast  iron 

Brass  upon  wrought  iron,  the   fibres  of) 

the  iron  being  parallel  to  the  motion  •  \ 

Wrought  iron  upon  brass 

Brass  upon  brass 

Black  leather  (curried)  upon  oak*    . 
Ox  hide  (such  as  that  used  for  soles  and 

for  the  stuffing  of  pistons)  upon  oak, 

roueh    

Ditto        ditto        ditto    smooth     . 
Leather  as  above,  polished  and  hardened  | 

by  hammering f 

Hempen   girth,  or  pulley-band,  (saiigle'] 

de  chanvre,)  upon  oak,  the  fibres  of 

the  wood  and  the  direction  of  the  cord 

being  parallel  to  the  motion '   • 
Hempen    matting,    woven    witli    small  / 

cords,  ditto.     "■ i 

Old  cordage,  IJ  incli  in  diameter,  ditto+ 


0,256 

0,232 

0,194 

o,i38 
0,490 

0,193 

0,132 

o.3i4 
0,147 

0,372 

0,394 

0,436 

0,202 
0,1 52 
0,189 
0,617 
0.217 

0,161 

0,172 
0,201 
0.265 

0,32 

0,335 
0,296 

0,52 

0,32 

0,32 


Friction  of 

(illESCKNCE. 


14°  22' 

14  9 

10  39 

7  52 
26  7 

11  3 

8  39 

17  26 

8  22 


21  3i 

23  34 

1 1  26 

8  39 

10  49 
3i  41 

12  i5 

9  9 

9  46 

11  22 
14  5i 

27  29 

iS  3i 

16  3o 


27     29 

17    45 
27     29 


0,649 

0,194 
0,137 

0,646 
0,162 


0,617 

0,74 

o,6o5 

0,43 

0,64 

o,5o 
0,79 


33°    o' 
10    59 

7     49 

32     5j 
9     i3 


3i  41 

36  3 1 

3 1  II 

23  17 

32  33 

26  34 

38  19 


•  The  fiiclion  of  iiiolion  w:>s  very  ncnrlv  the  samo  whctliT  tliB  surfico  of  coiitict  was  the  inside 
•.ir  the  oiitsiile  of  the  skin. — The  cnnslnvcy  of  the  coclTicient  of  ilic  friclioa  of  motion  was  cqu:\ny  ap- 
parent in  the  roiij-h  and  the  smooth  skins. 

t  All  the  above  experiments,  except  th:U  witli  curried  blnck  leather,  presented  the  iilienomenon  of 
a  channe  in  the  polish  of  the  surfaces  of  friction— a  state  of  their  snrfares  necessary  to,  and  dependent 
ai>on.  their  motion  upon  one  another. 


TABLE   VI. 

TABLE  Yl—contk/ued. 


499 


Friction  of 

1 
Friction  of         1 

SURFACES  OF  CONTACT. 

Motion. 

QUIE 

SCENCE 

_• 

QJ 

*-•  c 

d 

£•- 

tiHs  = 

S.2 

615.- 

c 

^  eC 

c   C 

y  -^ 

05  ♦^ 

u  u 

i££ 

■=■ 

i'« 

*!£ 

=  ■5.-^        1 

O  c 

'2<.s.. 

0^ 

5< 

« 

Calcareous  oolitic  stone,  used  in  building, "] 

of  a  moderately  hard  qualitj',  called  1 
stone   of    Jauniont — upon    the  same  | 

0,64 

32' 

38' 

0,74 

36" 

3i' 

teiono      ••••■•••••! 
Hard  calcareous   stone  of  Brouck,  of  a] 

light  gray  color,  susceptible  of  taking  1 
a  fine  polish,  (the  muschelkalk,)  niov-  [ 

0,38 

20 

49 

0,70 

35 

0 

iug  upon  the  same  stone j 

The  soft  stone  mentioned  above,  upon  ^ 
the  hard ) 

0,65 

33 

2 

0,75 

36 

53 

The  hard  stone  mentioned  abo\e  upon 
the  soft 

0,67 

33 

5o 

0,75 

36 

53 

Common  brick  upon  the  stone  of  Jaumont 

0,65 

33 

2 

0,65 

33 

2 

Oak  upon  ditto,  the  fibres  of  the  wood  1 

being  perpendicular  to  the  surface  of  >- 

0,38 

20 

49 

0,63 

32 

i3 

the  stone ) 

Wrought  iron  upon  ditto,  ditto    •     •     • 

0,69 

34 

37 

Oj49 

26 

7 

Common  brick  upon  the  stone  of  Brouck 

0,60 

3o 

58 

0,67 

33 

DO 

Oak  as  before  (endwise)  upon  ditto  •     • 

0.38 

20 

49 

0,64 

32 

38 

Iron,                     ditto            ditto       •     • 

0,24 

i3 

3o 

0,42 

22 

47 

'/// 


500 


TABLE  Y  1 1. 


TABLE  YIL 

EXPEEIMEXTS  ON  THE  FKICTION  OF  UNCTUOUS  SURFACES. 
BY  M.  MORIN.— See  §  255. 

Ill  these  experiments  the   surfaces,  iifter  liaviniT  been  smeared  with  an   nnguenl,  wers 
wiped,  so  tliat  no  interposing  layer  of  the  unguent  prevented  their  intimate  contact. 


Frk 

riON  0 

F 

Friction  of          j 

SURFACES  OF   CONTACT. 

Motion. 

auiE 

SCENC 

E. 

C 

"t! 

U 

<2S 

Oak  upon  oak,  the  fibres  being  parallel  to  (_ 
the  motion f 

0,1 08 

6° 

iO' 

0,390 

21°     19' 

Ditto,  the  fibres  of  the  moving  body  be-  ( 
ing  perpendicular  to  the  motion-          •  f 

0,143 

8 

9 

o,3i4 

17 

26 

Oak  upon  ehii,  tiljres  parallel- 

O.I  36 

7 

45 

Elm  upon  oak,  ditto      .... 

0. 1 19 

6 

48 

0,420 

22 

47 

lieech  upon  oak,  ditto   .... 

o,33o 

18 

16 

Elm  upon  elm,  ditto      .... 

0. 140 

7 

59 

Wrought  iron  njion  elm,  ditto 

o.i38 

7 

32 

Ditto  upon  wrought  iron,  ditto 

0,177 

10 

3 

Ditto  upon  cast  iron,  tlitto     • 

0,118 

6 

44 

Cast  iron  upon  wrought  iron,  ditto 

0.143 

8 

9 

Wrought  iron  upon  brass,  ditto     • 

0. 1 60 

9 

6 

Brass  upon  wrought  iron 

0,166 

9 

26 

Cast  iron  upon  oak,  ditto 

0,107 

6 

7 

0,100 

5 

43 

Ditto  upon  elm,  ditto,  the  unguent  being  | 
tallow         •          •          •         "         •         •  f 

0.120 

7 

8 

Ditto,   ditto,  the   unguent  being   hog's  i_ 
lard  and  black  lead     •          •          •          •  ) 

0,137 

7 

49 

Elm  upon  cast  iron,  tibres  parallel  • 

0,1 35 

7 

42 

0,098 

5 

36 

Cast  ii-on  upon  cast  iron 

0,144 

8 

12 

Ditto  upon  brass 

0.l32 

7 

32 

Brass  upon  cast  iron     .         .         .         • 

0,107 

6 

7 

Ditto  upon  brass 

0,1 34 

7 

3S 

0,164 

9 

19 

Copper  upon  oak  .          .          .          •          • 

0.100 

5 

43 

Yellow  copper  upon  cast  iron 

0,1 15 

6 

34 

Leather  (ox  hide)  well  tanned  upon  cast  ( 
iron,  wetted ) 

0,229 

12 

54 

0,267 

14 

57     1 

Ditto  upon  bra^s,  wetted 

0,244 

i3 

43 

1 
1 
J 

TABLE    VIII. 

TABLE  VIII. 


501 


EXPERIMENTS  ON  FRICTION  WITH  UNGUENTS  INTERPOSED.    BY  M.  IIORIN. 

Tiie  extent  of  the  surfaces  in  tliese  experiments  bore  such  a  relation  to  the  pressure,  as 
lo  c-iuise  them  to  be  separated  from  one  anotlier  throughout  by  an  interposed  stratum  of 
tlic  unguent.— See  §355. 


Friction 

FnicTioN 

OF 

OF 

SURFACES  OF  CONTACT. 

Motion. 

(iuiESCKNCE. 

UNGUENTS. 

_, 

^ 

5      = 

2    = 

^       'Z, 

P     t. 

-     fe. 

o 

u 

Oak  upon  oak,  fibres  parallel 

0,164 

0,440 

P/rv  s<n.p. 

Ditto         ditto 

0,075 

0,164 

Tallow. 

Ditto         ditto 

0,067 

ling-.,  hud. 

Ditto,  fibres  perpendicular 

o,o83 

0,234 

Tali'.Nv. 

Ditto         ditto 

0,072 

11. .--s  lard. 

Ditto        ditto 

0.230 

Wat  IT. 

Ditto  upon  elm,  fibres  parallel 

0,1 36 

Di-\  ^oap. 

Ditto        ditto 

0.073 

0,178 

Tallow. 

Ditto        ditto 

0,066 

lioir's  lard. 

Ditto  upon  cast  iron,  ditto 

0,080 

'l'ario\\. 

Ditto  upon  wrought  iron,  ditto 

0.098 

Tallow. 

Beech  upon  oak,  ditto 

o,o55 

Tallow. 

Elm  upon  oak,  ditto  • 

0.137 

0.41 1 

Drv  soap. 

Ditto        ditto 

0.070 

0,142 

Tailow. 

Ditto        ditto 

0.060 

Hog's  lard. 

Ditto  upon  elm,  ditto 

0.139 

0,217 

Drv  soap. 

/        Ditto  upon  cast  iron,  ditto 

0,066 

Tiiilow. 
i  Greased,  and 

Wrought  iron  upon  oak-,  ditto    • 

0,256 

0,649 

•^  saturated  with 
(  water. 

Ditto        ditto         ditto  . 

0,214 

Dry  soap. 

Ditto         ilitto        ditto  • 

o,o85 

0,108 

Tailow. 

Ditto  upon  elm,      ditto  • 

0.078 

Tallow. 

Ditto         ditto        ditto  • 

0,076 

llog's  lard. 

Ditto         ditto        ditto  - 

o,o55 

Olive  oil. 

Ditto  upon  Cast  iron,  ditto 

o,io3 

Tallow. 

Ditto        ditto        ditto  • 

0.076 

Ilog's  lard. 

Ditto        ditto        ditto  • 

0,066 

0,100 

Olive  oil. 

Ditto  upon  wrouEcht  iron,  ditto 

0,082 

Tallow. 

iJitto        ditto     ^  ditto  • 

o.o8i 

Host's  lard. 

Ditto        ditto         ditto  • 

0,070 

0,1 15 

Olive  oil. 

"Wrought  iron  upon  brass,  fibres  ) 

0,  io3 

Tallow. 

parallel ) 

Ditto         ditto         ditto  • 

0,075 

Ilog's  lard. 

Ditto         ditto         ditto  • 

0.078 

Olive  oil. 

Cast  iron  upon  oak,  ditto  • 

0,189 

Dry  soap. 
i  Greased,  and 

Ditto        ditto        ditto  • 

0,218 

0,646 

-|  saturated  witli 
1  water. 

Ditto        ditto         ditto  • 

0.078 

0,100 

Tallow. 

Ditto        ditto        ditto  • 

0,075 

Ilog's  lard. 

Ditto        ditto        ditto  • 

0,075 

0,100 

Olive  oil. 

Ditto  upon  elm,     ditto  • 

0.077 

Tallow. 

Ditt-o        ditto         ditto  • 

0.061 

Olive  oil. 

DittD        ditto        ditto  • 

o,ogi 

j  Ilog's  lard  and 
1  jilunibago. 

Ditto,  ditto  upon  wrought  iron 

0,100 

Tallow. 

Cast  iron  upon  cast  iron     • 

o,3i4 

Water. 

Ditto         ditto 

0.197 

Soap. 

502 


TABLE    VIII. 


TABLE  YllL—contmued. 


Friction 

Friction 

OF 

OK 

SURFACES  OF  CONTACT. 

Motion. 

CiUIESCKNCE. 

UNGUENTS. 

^ 

^ 

"5.     ° 

9i     ''^ 

- 

S  "^ 

a  ^ 

Cast  iron  upon  cast  iron     • 

0,100 

0,100 

Tallow. 

<  Ditto         ditto 

0,070 

0,100 

Hogs'  lard. 

Ditto    "    ditto 

0,064 

. 

Olive  oil. 

"     Ditto        ditto 

o,o55 

Lard  and 
plumbago. 

Ditto  upon  brass    • 

o,io3 

Tallow. 

Ditto         ditto 

0,075 

Hogs'  hn-d. 

Ditto        ditto 

0,078 

Olive  oil. 

Copper  upon  oak,  fibres  paralle 
Yellow  copper  npou  cast  iron 

0,069 

0,100 

Tallow. 

0,072 

o,io3 

Tallow. 

Ditto         ditto 

0,068 

Hogs'  lard. 

Ditto        ditto 

0,066 

Olive  oil. 

Brass  upon  cast  iron- 

0,086 

0,106 

Tallow. 

Ditto         ditto 

0,077 

Olive  oil. 

Ditto  upon  wrought  iron 

0,081 

Tallow. 

Ditto        ditto 

0,089 

]a\i(]  and 
plumbago. 

Ditto        ditto 

0,072 

Olive  oil. 

Ditto  upon  brass     • 

o,o58 

Olive  oil. 

Steel  upon  east  iron   • 

o.ioS 

0,108 

Tallow. 

Ditto        ditto 

0,081 

lIog>'  lard. 

Ditto         ditto 

0,079 

Olivo  oil. 

Ditto  upon  wrouglit  iron 

0.093 

Tallow. 

Ditto        ditto 

0,076 

Hogs'  lard. 

Ditto  upon  brass     • 

o,o56 

Tallow. 

Ditto        ditto 

o,o53 

Olive  oil. 

Ditto        ditto 

0,067 

J  Lnrd  and 
1  )ilumbago. 

(  Greasecl,  and 

Tanned  ox  hide  upon  cast  iron 

0,365 

-|  saturated  with 
f  water. 

Ditto         ditto 

0,1 59 

Tallow. 

Ditto         ditto 

0,1 33 

0,122 

Olive  oil. 

Ditto  upon  brass    • 

0,241 

Tallow. 

Ditto        ditto 

0,191 

■ 

Olive  oil. 

Ditto  upon  oak, 

0,29 

0,70 

Water. 

Hempen  fibres  not  twisted,  mov-" 

ing  upon  oak,  tlie  fibres  of  the 
hemp  being  phiced  in  a  direc- 
tion ]ierpendiciilar  to  tlie  dircc^  ( 

0.332 

0,869 

(  Greased,  and     1 
■<  satiu'atecl  with 
(  water. 

tion  of  ilie  niDtion,  and  tliose 

of  tlic  oalc  fiarallel  to  it  • 

Tlic  same  as  above,  moving  upon  i 

0,194 

Tallow. 

cast  iron         •          •          •          •  ) 

Ditto 

0, 1 53 

Olive  oil. 

Soft  caleai-coiis  stone  of  Jaumonl" 
upon  the  sanu-,  witii  a  layer  of 

niortar,  of  sand,  and  lime  inter-  ► 

0,74 

posed,  after  from  10  to  15  min- 

utes' cmtact. 

TABLE    IX. 


503 


TABLE    IX. 

FEICTION   OF  TRUNNIONS   IN   THEIE   BOXES.— Sec  g  3G1. 


Ratio    of 

friction    to 

pressure 

when    the 

KINDS  OF  MATERIALS. 

STATE  OF  SURFACES. 

unguent 

s  renewed. 

By  ihB 

Or,  con- 

on  1  ill  HIT 

tin  (hiusly. 

nietliod. 

• 

r 

Unguents  of  olive  oil,  hogs'  lard, 

i   "I'o^  i 

o,o54 

and  tallow     •          •         •    .'     • 

1  o',o8  y 

Trunnions  of  cast  iron  and  J 

The  same  unguents  moistened  with 

water 

o,o8 

o,o54 
o,o54 

boxes  of  cast  iron. 

Unguent  of  asphaltnm 

o,o54 

Unctuous 

0,14 

Unctuous  and  moistened  witli  wa- 

ter          

o.U 

Unguents  of  olive  oil,  hogs'   lavd, 
and  tallow      .         .         .         . 

1  0,07    1 

•       to      } 

1  0.08  ) 
0,16 

o,o54 

Trunnions  of  cast  iron  and 
boxes  of  brass. 

Unctuous 

Unctuous  and  moistened  with  wa- 

ter           

0,16 

I 

Very  sliglitly  inietuous 

0,19 

Without  unguents     • 

0,18 

Uniruents  of  olive  oil  and  hogs'  f 

Trunnions  of  east  iron  and 
boxes  of  lignum-vitaj. 

lard        •          •          .          ■      "  ■  \ 

0,090 

Unctuous  with  oil  and  hogs'  lard 

0,10 

Unctuous  with  a  mixture  of  hogs" 

lard  and  pluir.bago 

0,14 

Trunnions  of  wrongbt  iron  _ 
and  boxes  of  cast  iron. 

Unguents  of  olive  oil,  tallow,  and 
hogs'  lard       .... 

[  0,07  1 
I  0,08  J 

o,o54 

> 

Trunnions  of  wrought  iron  ^ 
and  boxes  of  brass. 

Unguents   of  olive  oil,  hogs'  lard, 
and  tallow       .... 

Old  uniruents  hardened     • 

0.07  ) 
to 

( 0.08  ) 

0,09 

o.o54 

Unctuous  and  moistened  with  wa- 

ter         •          •          •          •          • 

0,19 

Very  slightly  unctuous 

0,23 

Trunnions  of  wrought  iron  | 

Unguents  of  oil  (jr  hogs'  lard     • 

0,11 

and  boxes  of  lignuni-vi-  } 

Unctuous 

0,19 

Trunnions    of  brass   and  ( 

Unguent  of  oil- 

O.IO 

- 

boxes  of  brass.                 1 

Unguent  of  hog.--'  lard 

0.09 

Trunnions   of  brass   and  | 

Unguents  of  tallow  or  of  olive  oil. 

( 0,045  1 

boxes  of  cast  iron.            j 

( o.o52  ) 

Trunnions  of  lignuni-vitLc  ( 

Unguents  of  hogs'  lard 

0,12 

1 

and  boxes  of  cast  iron.    ) 

Unctuous-          .... 

o,i5 

. 

Trunnions  of  lignum-vitre  i 

and   boxes   of   lignum-  V 

Unguent  of  hogs'  lard 

• 

0,07 

vitiie.                                  ) 

504 


TABLE     X. 


TABLE     X. 

OF  WEIGHTS  NECESSAEY  TO  BEND  DIFFERENT  EOPES  AEOUND  A  WHEEL 
ONE  FOOT  IN  DIAMKTEE.— See  §357 

No.  1.    White  Eopks — nkw  and  miY. 
Stiffnesb  proportional  to  the  square  of  the  iiameler. 


Diameter  of  rope 
in  iiioii  s. 

Nalnml  stiirnes*, 
or  Viilue  of  K. 

Stiffness  for  load  of 
J  lb.,  or  value  of/. 

0,39 
0-79 

1,37 

3,i5 

lbs. 

0,4024 

1,6097 

6,4089 

25,7553 

lbs. 

0,0079877 
o,o3i93oi 
0,1278019        ] 

0,3112019            ! 

No,  2.    White  Bopes — new  and  moistened    with 

WATEl'u. 


Stiffness  proportional  to  square  of  diameter. 

Diniiicter  of  rope 
in  inches. 

N.iturtil  slitriiess,    jSliffness  f.rr  IikkI  oI 
or  vnhie  of  K.        1  11).,  or  value  of  I. 

0,39 
0,79 
«,37 
3,i5 

lbs. 

0,8048 

3,2194 

12.8772 

5i,5iii 

lbs. 

0,0079877 
o,o3i95oi 
0,1278019 
0,5112019 

Squares  of  the  ratios 

ol  iliaiiieter,  or  Val- 

ues of  cp^ 

Ratios  (/. 

^■qn•lres 

(/-. 

1,00 

1,00 

1,10 

1. 21 

1,20 

1,44 

l,3o 

1,69 

1, 40 

1,96 

1,30 

2,23 

1,60 

2.56 

1,70 

2  69 

1,80 

3.24 

1,90 

3.61 

2,00 

4,00 

No.  3.    White  Eopes — half  -nrouN  and  dhy. 

Stiffness  proportional  to  the  square  root  of  the  cube  of 
the  diameter . 


Diameter  of  rojie 
in  inches. 

Xalnral  SiillV.fss, 
or  value  of  A". 

Slillriess  forloail  ofj 
1  11).,  or  value  of  7. 

0,39 

o>79 
1,37 

3,13 

lbs. 
0,40243 
i,i38oi 
3,21844 
9,ioi5o 

0,0079877 
0,0523889 
0,0638794 
0,1806373 

No.  4.   White  Eope.s — half  woun  and  moi-stexed 
WITH   watei;. 

Stiffness  proportional  to  the  square  root  <f  the  cuhe  of 
the  dlanuler. 


Diatneter  of  rope 
in  Inches. 

Natural  Sliffness, 
or  value  of  A-. 

Stllfiie^s  for  lo  1(1  of 
1  11).,  or  value  of/. 

0,39 
0.79 
1,37 

3,i5 

lbs. 

0,8048 

2.2761 

6.4324 

18,2087 

lbs. 

0,0079877 

0,0523889 

0,0638794 

0,1806373 

r^qii:ire  roots  of  the 

cubes  ot  the  r-itios 

ot  (lianietcr,  or  val 

3 

lies  of  (J 2' 

llatio?  or 
d. 

Power  ? 

'"■  d2- 

1. 00 

1,000 

1,10 

1. 1 54 

1,20 

i.3i5 

i.3o 

1.482 

1.40 

1.30 
1,60 

1.657 
1,837 
2,024 

I.IO 

2.217 

1.80 

2,4i5 

1,90 

,00 

2,619 
2,828 

APPENDIX.  505 

TABLE    X — continued. 

No.  5.  Tauked  Kopes. 

Stiffness  proportional  to  the  number  of  yarns. 

[These  ropes  are  usually  made  of  three  strands  twisted  around  each  other,  each  strand  being  com- 
posed of  a  certain  number  of  yarns,  also  twisted  abou'  each  other  in  the  same  manner.] 


A  P  P  E  N  D  I  X  . 

N...  J. 

Take  tlie  usual  fbrniiilas  ibr  the  traiisformatiou  of  co-ordiiiales  from 
one    system   to    another,    holh    being  rectangular,  viz : 

X  =  a  x'  +  (^ '/'  +  c  z' ,      "I 

y  =  a  X   +b  y   ^-  c  z  ,      )■ (1) 

z  =  a".i'  +6"i/'  -j-  c"z' ;  j 

in  Y/hich  a,  h,  etc.,  denote  the  cosine.^  of  the  angles  which  ihe  axes  of 
the  same  name  as  the  co-ordinal es  into  wliieh  they  are  respectively 
multiplied  make  with  the  axis  of  the  variable  in  the  first  member 
And    hence, 

x'  =  a  X  -\-  a   >i  +  «"  ■J'-    1 

y'  =.hx  +  b'  y  +  I/'  z,     \ (2) 

z    —  c  X  +  c  y  +  c    z  ;  j 

Multiply  the    first    of  (2)  by  b,  the    second    by  a,  and    take  tlie  dil- 
ference  of  the  products;   we  get 

b  x'  —  a  if  =  y  {a  b  —  ab')  -\-  z  [a"  b  —  a  b") ;  •  •  •  (3) 
again,  m.ultiply  the  first  by  c,  the  third  by  a,  and  take  the  diliereuci^ 
of  products ;    we    have 

c  x'  —  a  z'  z^  y  {a'  c  —  a  c')  -{-  z  [a"  c  —  a  c")         •     •     (1) 
Find  the    value    of  //  in  (4),  substitute    in    (3),   and  reduce,  Ave  rui'.l 
Az  =  {b  c'  —  b'  c)  x'  -f  («'  c  —  a  c')  y'  +  (a  b'  —  a'  b)  z' , 


506  APPENDIX. 

in   which 

A  =  c  {a'  h"  -  a"  b')  +  c'  {a"  h-a  h")  +  c"  {a  U  -  a'  b) , 

dividing    by  A,  and    subtracliiig    tlie    result  irom  the  lhird    of  Eqs.  (1) 
we   have 


/   „       be  —  b' c\    ,        /  a'c  —  ac'\    ,       /  ,,       ab'  —a' b\    , 

and   since    x' ,  y'  and  z'  are  wholly  arbitrary,  we  have 

a"  -  ^Jl^  =0;b"-  ''''-/'''  =.  0  ;  ."  -  'll^/A  =.  o ;  •  (5) 
A  A  A  '     \  J 

transposing,    clearing    the    traction,    squaring,    adding,    collecting    the    co- 
efficients of'f'^,  b'",  a'",  and  reducing    by  the    relations 

a~  +  b-  -\-  C"  —  1  ;  a""  +  b''^  +  c'-  :=  1 ; 

«"  +  6-  =  1  —  c~ ;  c~  +  ^-  =z  1  —  a~ ;  a-  +  c-  =  1  —  b^, 
there    will    result 

A2=    1    -(««/   +   Z*i'   -\-Cc'f. 

But 

«  a'  +  'I'  <^'  +  c  c'  =  0, 

whence    A  =z  \,  and,  Eqs.  (o), 

a"  ^  b  cf  —  b'  c ;  Z*"  ;=  a  c  —  a  c  ;  c"  z::i  a  b'  -  -  a'  b. 

No.  II. 
To    find    the    radius    of  curvature    of  any    curve,   and    its    inclination 

to    the    co-ordinate    axes. 

Take    the    centre   of  curvature    as    the  centre  of  a  sphere  of  which 

the   radius   is   unity.      Through    the    same 

point    draw  the    line   O  X,  parallel    to    the 

axis  X,   and    another   O   T,    parallel  1o    the 

tangent    to    the    arc    M  iV,  of    osculation. 

The    planes    of  tbese  lines  and  of    the  ra- 
dius of  curvature  will  cut  from    the  sphere 

the  spherical  triangle  A  B  C,  of  which  the 

side    B  C  is  90°,   A  C  the  angle  which  the  radius  of  curvature  makes 

with  the  axis   .r,   and  A  B    the  angle  which  the  tangent  to    the    curve 

makes    with  the    same    axis.      Make 

p  =  OR—  radius    of  curvature, 
/)'=  yl  C:  r  =  AB:   C  =  A  C  B. 


h-X 


/y 


APPENDIX.  507 

Then   will 

d  % 
■^  cos  c  =^  -7-^  =^  sin  S'.  cos  C  ; 
a  s 

diflerentiatiiig,    and   regarding    C   as   constant, 

a  -r-  — cos  6  .  (16  .  cos  C  : 
a  s 

but  d6'.cosC    h    the    projection    of    the    arc    d 6'    on    the    osculatory 
plane,    whence 

d  &'.  cos  C  =:  —  • 
P 
Substituting   tliis  above,    we   find 

,  d  X 

.,  a  s 

cos  &     =z  a  •  • 

^         ds  ' 

and    denoting    by    6"    and    d'",    the     angles    which    the     radius    njakes 
MTth   the   axes  y   and   z,    respectively,  we    may  write 

,  dx  ,  d  1/                                 ,  d z 

d—-  d-r^                               d-r- 

,,                 a  s  ,,,                d  s                .,,,               d  s 

cos  a    =  p  •    ■      ■  ;  cosd     =zp- — —- ;  cost)      —0 —  •  •     •   (1) 

d  s  d  s                                   is 

S'fuaring,  adding   and   reducing    by    the   relation. 

cos3  (3'  +  cos-  6"  +  cos^  6'"  =  1, 
we    have 

d  s 


d  z\-^ ' 


perlbrming    the    operations    indicated   under    the    radical   sign,   and  reilu- 
cing   by   the   relations 

d  62  -  d  X'  +  dif  +  d  c2, 

fZ-  s  d  s  ^  d~  X  d  X  -\-  d"  y  d  y  -\-  d-  z  d  z, 
we    find 

ds^ 

^  ~~  ~-y/{d'' xf+'{d:^yf+~{d?  zf  —  (#1p'         *     *     ■  ^^^ 

If  s   be    taken    as    the    independent    variable,   then    will    d^  s  :=  0,    and 
Eqs.    (1)    and    (2)    become 

,,  d"  X  ,,,  d"^  7/  ^,,,  d~  z 

cos  6'    =  p  ■  — — •  ;    cosd"  =  p  •  —4  ;    cos^  "  =  p  •  — —  ;    •     •     (;j) 
d  S"  d  6-  d  A'"' 

d^ 


,/-- 


'^b 


2  9 


-^  t). 


/. 


A    ^>^^^  ^ 


/'^ 


r'l  e' 


t' 


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